diff --git a/src/HOL/Analysis/Elementary_Topology.thy b/src/HOL/Analysis/Elementary_Topology.thy
--- a/src/HOL/Analysis/Elementary_Topology.thy
+++ b/src/HOL/Analysis/Elementary_Topology.thy
@@ -1,2666 +1,2660 @@
 (*  Author:     L C Paulson, University of Cambridge
     Author:     Amine Chaieb, University of Cambridge
     Author:     Robert Himmelmann, TU Muenchen
     Author:     Brian Huffman, Portland State University
 *)
 
 chapter \<open>Topology\<close>
 
 theory Elementary_Topology
 imports
   "HOL-Library.Set_Idioms"
   "HOL-Library.Disjoint_Sets"
   Product_Vector
 begin
 
 
 section \<open>Elementary Topology\<close>
 
-subsection \<open>TODO: move?\<close>
-
-lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
-  using openI by auto
-
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Affine transformations of intervals\<close>
 
 lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c  \<longleftrightarrow> inverse m * y + - (c / m) = x"
   for m :: "'a::linordered_field"
   by (simp add: field_simps)
 
 
-
 subsection \<open>Topological Basis\<close>
 
 context topological_space
 begin
 
 definition\<^marker>\<open>tag important\<close> "topological_basis B \<longleftrightarrow>
   (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
 
 lemma topological_basis:
   "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
   unfolding topological_basis_def
   apply safe
      apply fastforce
     apply fastforce
    apply (erule_tac x=x in allE, simp)
    apply (rule_tac x="{x}" in exI, auto)
   done
 
 lemma topological_basis_iff:
   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
   shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
     (is "_ \<longleftrightarrow> ?rhs")
 proof safe
   fix O' and x::'a
   assume H: "topological_basis B" "open O'" "x \<in> O'"
   then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
   then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
   then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
 next
   assume H: ?rhs
   show "topological_basis B"
     using assms unfolding topological_basis_def
   proof safe
     fix O' :: "'a set"
     assume "open O'"
     with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
       by (force intro: bchoice simp: Bex_def)
     then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
       by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
   qed
 qed
 
 lemma topological_basisI:
   assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
     and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
   shows "topological_basis B"
   using assms by (subst topological_basis_iff) auto
 
 lemma topological_basisE:
   fixes O'
   assumes "topological_basis B"
     and "open O'"
     and "x \<in> O'"
   obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
 proof atomize_elim
   from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
     by (simp add: topological_basis_def)
   with topological_basis_iff assms
   show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
     using assms by (simp add: Bex_def)
 qed
 
 lemma topological_basis_open:
   assumes "topological_basis B"
     and "X \<in> B"
   shows "open X"
   using assms by (simp add: topological_basis_def)
 
 lemma topological_basis_imp_subbasis:
   assumes B: "topological_basis B"
   shows "open = generate_topology B"
 proof (intro ext iffI)
   fix S :: "'a set"
   assume "open S"
   with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
     unfolding topological_basis_def by blast
   then show "generate_topology B S"
     by (auto intro: generate_topology.intros dest: topological_basis_open)
 next
   fix S :: "'a set"
   assume "generate_topology B S"
   then show "open S"
     by induct (auto dest: topological_basis_open[OF B])
 qed
 
 lemma basis_dense:
   fixes B :: "'a set set"
     and f :: "'a set \<Rightarrow> 'a"
   assumes "topological_basis B"
     and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
   shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
 proof (intro allI impI)
   fix X :: "'a set"
   assume "open X" and "X \<noteq> {}"
   from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
   obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
   then show "\<exists>B'\<in>B. f B' \<in> X"
     by (auto intro!: choosefrom_basis)
 qed
 
 end
 
 lemma topological_basis_prod:
   assumes A: "topological_basis A"
     and B: "topological_basis B"
   shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
   unfolding topological_basis_def
 proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
   fix S :: "('a \<times> 'b) set"
   assume "open S"
   then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
   proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
     fix x y
     assume "(x, y) \<in> S"
     from open_prod_elim[OF \<open>open S\<close> this]
     obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
       by (metis mem_Sigma_iff)
     moreover
     from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
       by (rule topological_basisE)
     moreover
     from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
       by (rule topological_basisE)
     ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
       by (intro UN_I[of "(A0, B0)"]) auto
   qed auto
 qed (metis A B topological_basis_open open_Times)
 
 
 subsection \<open>Countable Basis\<close>
 
 locale\<^marker>\<open>tag important\<close> countable_basis = topological_space p for p::"'a set \<Rightarrow> bool" +
   fixes B :: "'a set set"
   assumes is_basis: "topological_basis B"
     and countable_basis: "countable B"
 begin
 
 lemma open_countable_basis_ex:
   assumes "p X"
   shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
   using assms countable_basis is_basis
   unfolding topological_basis_def by blast
 
 lemma open_countable_basisE:
   assumes "p X"
   obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
   using assms open_countable_basis_ex
   by atomize_elim simp
 
 lemma countable_dense_exists:
   "\<exists>D::'a set. countable D \<and> (\<forall>X. p X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
 proof -
   let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
   have "countable (?f ` B)" using countable_basis by simp
   with basis_dense[OF is_basis, of ?f] show ?thesis
     by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
 qed
 
 lemma countable_dense_setE:
   obtains D :: "'a set"
   where "countable D" "\<And>X. p X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
   using countable_dense_exists by blast
 
 end
 
 lemma countable_basis_openI: "countable_basis open B"
   if "countable B" "topological_basis B"
   using that
   by unfold_locales
     (simp_all add: topological_basis topological_space.topological_basis topological_space_axioms)
 
 lemma (in first_countable_topology) first_countable_basisE:
   fixes x :: 'a
   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
 proof -
   obtain \<A> where \<A>: "(\<forall>i::nat. x \<in> \<A> i \<and> open (\<A> i))" "(\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
     using first_countable_basis[of x] by metis
   show thesis
   proof 
     show "countable (range \<A>)"
       by simp
   qed (use \<A> in auto)
 qed
 
 lemma (in first_countable_topology) first_countable_basis_Int_stableE:
   obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
     "\<And>A B. A \<in> \<A> \<Longrightarrow> B \<in> \<A> \<Longrightarrow> A \<inter> B \<in> \<A>"
 proof atomize_elim
   obtain \<B> where \<B>:
     "countable \<B>"
     "\<And>B. B \<in> \<B> \<Longrightarrow> x \<in> B"
     "\<And>B. B \<in> \<B> \<Longrightarrow> open B"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>B\<in>\<B>. B \<subseteq> S"
     by (rule first_countable_basisE) blast
   define \<A> where [abs_def]:
     "\<A> = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into \<B> n) ` N)) ` (Collect finite::nat set set)"
   then show "\<exists>\<A>. countable \<A> \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> x \<in> A) \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> open A) \<and>
         (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)) \<and> (\<forall>A B. A \<in> \<A> \<longrightarrow> B \<in> \<A> \<longrightarrow> A \<inter> B \<in> \<A>)"
   proof (safe intro!: exI[where x=\<A>])
     show "countable \<A>"
       unfolding \<A>_def by (intro countable_image countable_Collect_finite)
     fix A
     assume "A \<in> \<A>"
     then show "x \<in> A" "open A"
       using \<B>(4)[OF open_UNIV] by (auto simp: \<A>_def intro: \<B> from_nat_into)
   next
     let ?int = "\<lambda>N. \<Inter>(from_nat_into \<B> ` N)"
     fix A B
     assume "A \<in> \<A>" "B \<in> \<A>"
     then obtain N M where "A = ?int N" "B = ?int M" "finite (N \<union> M)"
       by (auto simp: \<A>_def)
     then show "A \<inter> B \<in> \<A>"
       by (auto simp: \<A>_def intro!: image_eqI[where x="N \<union> M"])
   next
     fix S
     assume "open S" "x \<in> S"
     then obtain a where a: "a\<in>\<B>" "a \<subseteq> S" using \<B> by blast
     then show "\<exists>a\<in>\<A>. a \<subseteq> S" using a \<B>
       by (intro bexI[where x=a]) (auto simp: \<A>_def intro: image_eqI[where x="{to_nat_on \<B> a}"])
   qed
 qed
 
 lemma (in topological_space) first_countableI:
   assumes "countable \<A>"
     and 1: "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
     and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
   shows "\<exists>\<A>::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
 proof (safe intro!: exI[of _ "from_nat_into \<A>"])
   fix i
   have "\<A> \<noteq> {}" using 2[of UNIV] by auto
   show "x \<in> from_nat_into \<A> i" "open (from_nat_into \<A> i)"
     using range_from_nat_into_subset[OF \<open>\<A> \<noteq> {}\<close>] 1 by auto
 next
   fix S
   assume "open S" "x\<in>S" from 2[OF this]
   show "\<exists>i. from_nat_into \<A> i \<subseteq> S"
     using subset_range_from_nat_into[OF \<open>countable \<A>\<close>] by auto
 qed
 
 instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
 proof
   fix x :: "'a \<times> 'b"
   obtain \<A> where \<A>:
       "countable \<A>"
       "\<And>a. a \<in> \<A> \<Longrightarrow> fst x \<in> a"
       "\<And>a. a \<in> \<A> \<Longrightarrow> open a"
       "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>\<A>. a \<subseteq> S"
     by (rule first_countable_basisE[of "fst x"]) blast
   obtain B where B:
       "countable B"
       "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
       "\<And>a. a \<in> B \<Longrightarrow> open a"
       "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
     by (rule first_countable_basisE[of "snd x"]) blast
   show "\<exists>\<A>::nat \<Rightarrow> ('a \<times> 'b) set.
     (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
   proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B)"], safe)
     fix a b
     assume x: "a \<in> \<A>" "b \<in> B"
     show "x \<in> a \<times> b" 
       by (simp add: \<A>(2) B(2) mem_Times_iff x)
     show "open (a \<times> b)"
       by (simp add: \<A>(3) B(3) open_Times x)
   next
     fix S
     assume "open S" "x \<in> S"
     then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
       by (rule open_prod_elim)
     moreover
     from a'b' \<A>(4)[of a'] B(4)[of b']
     obtain a b where "a \<in> \<A>" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
       by auto
     ultimately
     show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B). a \<subseteq> S"
       by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
   qed (simp add: \<A> B)
 qed
 
 class second_countable_topology = topological_space +
   assumes ex_countable_subbasis:
     "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
 begin
 
 lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
 proof -
   from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
     by blast
   let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
 
   show ?thesis
   proof (intro exI conjI)
     show "countable ?B"
       by (intro countable_image countable_Collect_finite_subset B)
     {
       fix S
       assume "open S"
       then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
         unfolding B
       proof induct
         case UNIV
         show ?case by (intro exI[of _ "{{}}"]) simp
       next
         case (Int a b)
         then obtain x y where x: "a = \<Union>(Inter ` x)" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
           and y: "b = \<Union>(Inter ` y)" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
           by blast
         show ?case
           unfolding x y Int_UN_distrib2
           by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
       next
         case (UN K)
         then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = k" by auto
         then obtain k where
             "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> \<Union>(Inter ` (k ka)) = ka"
           unfolding bchoice_iff ..
         then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = \<Union>K"
           by (intro exI[of _ "\<Union>(k ` K)"]) auto
       next
         case (Basis S)
         then show ?case
           by (intro exI[of _ "{{S}}"]) auto
       qed
       then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
         unfolding subset_image_iff by blast }
     then show "topological_basis ?B"
       unfolding topological_basis_def
       by (safe intro!: open_Inter)
          (simp_all add: B generate_topology.Basis subset_eq)
   qed
 qed
 
 
 end
 
 lemma univ_second_countable:
   obtains \<B> :: "'a::second_countable_topology set set"
   where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
        "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
 by (metis ex_countable_basis topological_basis_def)
 
 proposition Lindelof:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> open S"
   obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
 proof -
   obtain \<B> :: "'a set set"
     where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
       and \<B>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
     using univ_second_countable by blast
   define \<D> where "\<D> \<equiv> {S. S \<in> \<B> \<and> (\<exists>U. U \<in> \<F> \<and> S \<subseteq> U)}"
   have "countable \<D>"
     apply (rule countable_subset [OF _ \<open>countable \<B>\<close>])
     apply (force simp: \<D>_def)
     done
   have "\<And>S. \<exists>U. S \<in> \<D> \<longrightarrow> U \<in> \<F> \<and> S \<subseteq> U"
     by (simp add: \<D>_def)
   then obtain G where G: "\<And>S. S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"
     by metis
   have "\<Union>\<F> \<subseteq> \<Union>\<D>"
     unfolding \<D>_def by (blast dest: \<F> \<B>)
   moreover have "\<Union>\<D> \<subseteq> \<Union>\<F>"
     using \<D>_def by blast
   ultimately have eq1: "\<Union>\<F> = \<Union>\<D>" ..
   have eq2: "\<Union>\<D> = \<Union> (G ` \<D>)"
     using G eq1 by auto
   show ?thesis
     apply (rule_tac \<F>' = "G ` \<D>" in that)
     using G \<open>countable \<D>\<close>
     by (auto simp: eq1 eq2)
 qed
 
 lemma countable_disjoint_open_subsets:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" and pw: "pairwise disjnt \<F>"
     shows "countable \<F>"
 proof -
   obtain \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
     by (meson assms Lindelof)
   with pw have "\<F> \<subseteq> insert {} \<F>'"
     by (fastforce simp add: pairwise_def disjnt_iff)
   then show ?thesis
     by (simp add: \<open>countable \<F>'\<close> countable_subset)
 qed
 
 sublocale second_countable_topology <
   countable_basis "open" "SOME B. countable B \<and> topological_basis B"
   using someI_ex[OF ex_countable_basis]
   by unfold_locales safe
 
 
 instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
 proof
   obtain A :: "'a set set" where "countable A" "topological_basis A"
     using ex_countable_basis by auto
   moreover
   obtain B :: "'b set set" where "countable B" "topological_basis B"
     using ex_countable_basis by auto
   ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
     by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
       topological_basis_imp_subbasis)
 qed
 
 instance second_countable_topology \<subseteq> first_countable_topology
 proof
   fix x :: 'a
   define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
   then have B: "countable B" "topological_basis B"
     using countable_basis is_basis
     by (auto simp: countable_basis is_basis)
   then show "\<exists>A::nat \<Rightarrow> 'a set.
     (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
     by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
        (fastforce simp: topological_space_class.topological_basis_def)+
 qed
 
 instance nat :: second_countable_topology
 proof
   show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
     by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
 qed
 
 lemma countable_separating_set_linorder1:
   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
 proof -
   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
   define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
   proof (cases)
     assume "\<exists>z. x < z \<and> z < y"
     then obtain z where z: "x < z \<and> z < y" by auto
     define U where "U = {x<..<y}"
     then have "open U" by simp
     moreover have "z \<in> U" using z U_def by simp
     ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     define w where "w = (SOME x. x \<in> V)"
     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
     then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
     ultimately show ?thesis by auto
   next
     assume "\<not>(\<exists>z. x < z \<and> z < y)"
     then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
     define U where "U = {x<..}"
     then have "open U" by simp
     moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
     ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
     then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
     then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
     then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
     moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
     ultimately show ?thesis by auto
   qed
   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
   ultimately show ?thesis by auto
 qed
 
 lemma countable_separating_set_linorder2:
   shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
 proof -
   obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
   define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
   then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
   then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
   have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
   proof (cases)
     assume "\<exists>z. x < z \<and> z < y"
     then obtain z where z: "x < z \<and> z < y" by auto
     define U where "U = {x<..<y}"
     then have "open U" by simp
     moreover have "z \<in> U" using z U_def by simp
     ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     define w where "w = (SOME x. x \<in> V)"
     then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
     then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
     moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
     ultimately show ?thesis by auto
   next
     assume "\<not>(\<exists>z. x < z \<and> z < y)"
     then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
     define U where "U = {..<y}"
     then have "open U" by simp
     moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
     ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
     have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
     then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
     then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
     then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
     moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
     ultimately show ?thesis by auto
   qed
   moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
   ultimately show ?thesis by auto
 qed
 
 lemma countable_separating_set_dense_linorder:
   shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
 proof -
   obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
     using countable_separating_set_linorder1 by auto
   have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
   proof -
     obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
     then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
     then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
     then show ?thesis using \<open>b \<in> B\<close> by auto
   qed
   then show ?thesis using B(1) by auto
 qed
 
 
 subsection \<open>Polish spaces\<close>
 
 text \<open>Textbooks define Polish spaces as completely metrizable.
   We assume the topology to be complete for a given metric.\<close>
 
 class polish_space = complete_space + second_countable_topology
 
 
 subsection \<open>Limit Points\<close>
 
 definition\<^marker>\<open>tag important\<close> (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
   where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
 
 lemma islimptI:
   assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
   shows "x islimpt S"
   using assms unfolding islimpt_def by auto
 
 lemma islimptE:
   assumes "x islimpt S" and "x \<in> T" and "open T"
   obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
   using assms unfolding islimpt_def by auto
 
 lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
   unfolding islimpt_def eventually_at_topological by auto
 
 lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
   unfolding islimpt_def by fast
 
 lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
   unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
 
 lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
   unfolding islimpt_def by blast
 
 text \<open>A perfect space has no isolated points.\<close>
 
 lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
   for x :: "'a::perfect_space"
   unfolding islimpt_UNIV_iff by (rule not_open_singleton)
 
 lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
   unfolding closed_def
   apply (subst open_subopen)
   apply (simp add: islimpt_def subset_eq)
   apply (metis ComplE ComplI)
   done
 
 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   by (auto simp: islimpt_def)
 
 lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
   by (simp add: islimpt_iff_eventually eventually_conj_iff)
 
 
 lemma islimpt_insert:
   fixes x :: "'a::t1_space"
   shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
 proof
   assume *: "x islimpt (insert a s)"
   show "x islimpt s"
   proof (rule islimptI)
     fix t
     assume t: "x \<in> t" "open t"
     show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
     proof (cases "x = a")
       case True
       obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
         using * t by (rule islimptE)
       with \<open>x = a\<close> show ?thesis by auto
     next
       case False
       with t have t': "x \<in> t - {a}" "open (t - {a})"
         by (simp_all add: open_Diff)
       obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
         using * t' by (rule islimptE)
       then show ?thesis by auto
     qed
   qed
 next
   assume "x islimpt s"
   then show "x islimpt (insert a s)"
     by (rule islimpt_subset) auto
 qed
 
 lemma islimpt_finite:
   fixes x :: "'a::t1_space"
   shows "finite s \<Longrightarrow> \<not> x islimpt s"
   by (induct set: finite) (simp_all add: islimpt_insert)
 
 lemma islimpt_Un_finite:
   fixes x :: "'a::t1_space"
   shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
   by (simp add: islimpt_Un islimpt_finite)
 
 lemma islimpt_eq_acc_point:
   fixes l :: "'a :: t1_space"
   shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
 proof (safe intro!: islimptI)
   fix U
   assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
   then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
     by (auto intro: finite_imp_closed)
   then show False
     by (rule islimptE) auto
 next
   fix T
   assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
   then have "infinite (T \<inter> S - {l})"
     by auto
   then have "\<exists>x. x \<in> (T \<inter> S - {l})"
     unfolding ex_in_conv by (intro notI) simp
   then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
     by auto
 qed
 
 lemma acc_point_range_imp_convergent_subsequence:
   fixes l :: "'a :: first_countable_topology"
   assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
 proof -
   from countable_basis_at_decseq[of l]
   obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. l \<in> A i"
       "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
   {
     fix n i
     have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
       using l A by auto
     then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
       unfolding ex_in_conv by (intro notI) simp
     then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
       by auto
     then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
       by (auto simp: not_le)
     then have "i < s n i" "f (s n i) \<in> A (Suc n)"
       unfolding s_def by (auto intro: someI2_ex)
   }
   note s = this
   define r where "r = rec_nat (s 0 0) s"
   have "strict_mono r"
     by (auto simp: r_def s strict_mono_Suc_iff)
   moreover
   have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "l \<in> S"
     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
       by auto
     moreover
     {
       fix i
       assume "Suc 0 \<le> i"
       then have "f (r i) \<in> A i"
         by (cases i) (simp_all add: r_def s)
     }
     then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
       by (auto simp: eventually_sequentially)
     ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
       by eventually_elim auto
   qed
   ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
     by (auto simp: convergent_def comp_def)
 qed
 
 lemma islimpt_range_imp_convergent_subsequence:
   fixes l :: "'a :: {t1_space, first_countable_topology}"
   assumes l: "l islimpt (range f)"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
   using l unfolding islimpt_eq_acc_point
   by (rule acc_point_range_imp_convergent_subsequence)
 
 lemma sequence_unique_limpt:
   fixes f :: "nat \<Rightarrow> 'a::t2_space"
   assumes "(f \<longlongrightarrow> l) sequentially"
     and "l' islimpt (range f)"
   shows "l' = l"
 proof (rule ccontr)
   assume "l' \<noteq> l"
   obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
     using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
   have "eventually (\<lambda>n. f n \<in> t) sequentially"
     using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
   then obtain N where "\<forall>n\<ge>N. f n \<in> t"
     unfolding eventually_sequentially by auto
 
   have "UNIV = {..<N} \<union> {N..}"
     by auto
   then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
     using assms(2) by simp
   then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
     by (simp add: image_Un)
   then have "l' islimpt (f ` {N..})"
     by (simp add: islimpt_Un_finite)
   then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
     using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
   then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
     by auto
   with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
     by simp
   with \<open>s \<inter> t = {}\<close> show False
     by simp
 qed
 
 
 subsection \<open>Interior of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> interior :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
 
 lemma interiorI [intro?]:
   assumes "open T" and "x \<in> T" and "T \<subseteq> S"
   shows "x \<in> interior S"
   using assms unfolding interior_def by fast
 
 lemma interiorE [elim?]:
   assumes "x \<in> interior S"
   obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
   using assms unfolding interior_def by fast
 
 lemma open_interior [simp, intro]: "open (interior S)"
   by (simp add: interior_def open_Union)
 
 lemma interior_subset: "interior S \<subseteq> S"
   by (auto simp: interior_def)
 
 lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
   by (auto simp: interior_def)
 
 lemma interior_open: "open S \<Longrightarrow> interior S = S"
   by (intro equalityI interior_subset interior_maximal subset_refl)
 
 lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
   by (metis open_interior interior_open)
 
 lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
   by (metis interior_maximal interior_subset subset_trans)
 
 lemma interior_empty [simp]: "interior {} = {}"
   using open_empty by (rule interior_open)
 
 lemma interior_UNIV [simp]: "interior UNIV = UNIV"
   using open_UNIV by (rule interior_open)
 
 lemma interior_interior [simp]: "interior (interior S) = interior S"
   using open_interior by (rule interior_open)
 
 lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
   by (auto simp: interior_def)
 
 lemma interior_unique:
   assumes "T \<subseteq> S" and "open T"
   assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
   shows "interior S = T"
   by (intro equalityI assms interior_subset open_interior interior_maximal)
 
 lemma interior_singleton [simp]: "interior {a} = {}"
   for a :: "'a::perfect_space"
   apply (rule interior_unique, simp_all)
   using not_open_singleton subset_singletonD
   apply fastforce
   done
 
 lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
   by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
     Int_lower2 interior_maximal interior_subset open_Int open_interior)
 
 lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
   using interior_subset[of s] by (subst eventually_nhds) blast
 
 lemma interior_limit_point [intro]:
   fixes x :: "'a::perfect_space"
   assumes x: "x \<in> interior S"
   shows "x islimpt S"
   using x islimpt_UNIV [of x]
   unfolding interior_def islimpt_def
   apply (clarsimp, rename_tac T T')
   apply (drule_tac x="T \<inter> T'" in spec)
   apply (auto simp: open_Int)
   done
 
 lemma interior_closed_Un_empty_interior:
   assumes cS: "closed S"
     and iT: "interior T = {}"
   shows "interior (S \<union> T) = interior S"
 proof
   show "interior S \<subseteq> interior (S \<union> T)"
     by (rule interior_mono) (rule Un_upper1)
   show "interior (S \<union> T) \<subseteq> interior S"
   proof
     fix x
     assume "x \<in> interior (S \<union> T)"
     then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
     show "x \<in> interior S"
     proof (rule ccontr)
       assume "x \<notin> interior S"
       with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
         unfolding interior_def by fast
       from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
         by (rule open_Diff)
       from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
         by fast
       from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
         unfolding interior_def by fast
     qed
   qed
 qed
 
 lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
 proof (rule interior_unique)
   show "interior A \<times> interior B \<subseteq> A \<times> B"
     by (intro Sigma_mono interior_subset)
   show "open (interior A \<times> interior B)"
     by (intro open_Times open_interior)
   fix T
   assume "T \<subseteq> A \<times> B" and "open T"
   then show "T \<subseteq> interior A \<times> interior B"
   proof safe
     fix x y
     assume "(x, y) \<in> T"
     then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
       using \<open>open T\<close> unfolding open_prod_def by fast
     then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
       using \<open>T \<subseteq> A \<times> B\<close> by auto
     then show "x \<in> interior A" and "y \<in> interior B"
       by (auto intro: interiorI)
   qed
 qed
 
 lemma interior_Ici:
   fixes x :: "'a :: {dense_linorder,linorder_topology}"
   assumes "b < x"
   shows "interior {x ..} = {x <..}"
 proof (rule interior_unique)
   fix T
   assume "T \<subseteq> {x ..}" "open T"
   moreover have "x \<notin> T"
   proof
     assume "x \<in> T"
     obtain y where "y < x" "{y <.. x} \<subseteq> T"
       using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
     with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
       by (auto simp: subset_eq Ball_def)
     with \<open>T \<subseteq> {x ..}\<close> show False by auto
   qed
   ultimately show "T \<subseteq> {x <..}"
     by (auto simp: subset_eq less_le)
 qed auto
 
 lemma interior_Iic:
   fixes x :: "'a ::{dense_linorder,linorder_topology}"
   assumes "x < b"
   shows "interior {.. x} = {..< x}"
 proof (rule interior_unique)
   fix T
   assume "T \<subseteq> {.. x}" "open T"
   moreover have "x \<notin> T"
   proof
     assume "x \<in> T"
     obtain y where "x < y" "{x ..< y} \<subseteq> T"
       using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
     with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
       by (auto simp: subset_eq Ball_def less_le)
     with \<open>T \<subseteq> {.. x}\<close> show False by auto
   qed
   ultimately show "T \<subseteq> {..< x}"
     by (auto simp: subset_eq less_le)
 qed auto
 
 lemma countable_disjoint_nonempty_interior_subsets:
   fixes \<F> :: "'a::second_countable_topology set set"
   assumes pw: "pairwise disjnt \<F>" and int: "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"
   shows "countable \<F>"
 proof (rule countable_image_inj_on)
   have "disjoint (interior ` \<F>)"
     using pw by (simp add: disjoint_image_subset interior_subset)
   then show "countable (interior ` \<F>)"
     by (auto intro: countable_disjoint_open_subsets)
   show "inj_on interior \<F>"
     using pw apply (clarsimp simp: inj_on_def pairwise_def)
     apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)
     done
 qed
 
 subsection \<open>Closure of a Set\<close>
 
 definition\<^marker>\<open>tag important\<close> closure :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "closure S = S \<union> {x . x islimpt S}"
 
 lemma interior_closure: "interior S = - (closure (- S))"
   by (auto simp: interior_def closure_def islimpt_def)
 
 lemma closure_interior: "closure S = - interior (- S)"
   by (simp add: interior_closure)
 
 lemma closed_closure[simp, intro]: "closed (closure S)"
   by (simp add: closure_interior closed_Compl)
 
 lemma closure_subset: "S \<subseteq> closure S"
   by (simp add: closure_def)
 
 lemma closure_hull: "closure S = closed hull S"
   by (auto simp: hull_def closure_interior interior_def)
 
 lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
   unfolding closure_hull using closed_Inter by (rule hull_eq)
 
 lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
   by (simp only: closure_eq)
 
 lemma closure_closure [simp]: "closure (closure S) = closure S"
   unfolding closure_hull by (rule hull_hull)
 
 lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
   unfolding closure_hull by (rule hull_mono)
 
 lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
   unfolding closure_hull by (rule hull_minimal)
 
 lemma closure_unique:
   assumes "S \<subseteq> T"
     and "closed T"
     and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
   shows "closure S = T"
   using assms unfolding closure_hull by (rule hull_unique)
 
 lemma closure_empty [simp]: "closure {} = {}"
   using closed_empty by (rule closure_closed)
 
 lemma closure_UNIV [simp]: "closure UNIV = UNIV"
   using closed_UNIV by (rule closure_closed)
 
 lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
   by (simp add: closure_interior)
 
 lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
   using closure_empty closure_subset[of S] by blast
 
 lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
   using closure_eq[of S] closure_subset[of S] by simp
 
 lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
   using open_subset_interior[of S "- T"]
   using interior_subset[of "- T"]
   by (auto simp: closure_interior)
 
 lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
 proof
   fix x
   assume *: "open S" "x \<in> S \<inter> closure T"
   have "x islimpt (S \<inter> T)" if **: "x islimpt T"
   proof (rule islimptI)
     fix A
     assume "x \<in> A" "open A"
     with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
       by (simp_all add: open_Int)
     with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
       by (rule islimptE)
     then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
       by simp_all
     then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
   qed
   with * show "x \<in> closure (S \<inter> T)"
     unfolding closure_def by blast
 qed
 
 lemma closure_complement: "closure (- S) = - interior S"
   by (simp add: closure_interior)
 
 lemma interior_complement: "interior (- S) = - closure S"
   by (simp add: closure_interior)
 
 lemma interior_diff: "interior(S - T) = interior S - closure T"
   by (simp add: Diff_eq interior_complement)
 
 lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
 proof (rule closure_unique)
   show "A \<times> B \<subseteq> closure A \<times> closure B"
     by (intro Sigma_mono closure_subset)
   show "closed (closure A \<times> closure B)"
     by (intro closed_Times closed_closure)
   fix T
   assume "A \<times> B \<subseteq> T" and "closed T"
   then show "closure A \<times> closure B \<subseteq> T"
     apply (simp add: closed_def open_prod_def, clarify)
     apply (rule ccontr)
     apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
     apply (simp add: closure_interior interior_def)
     apply (drule_tac x=C in spec)
     apply (drule_tac x=D in spec, auto)
     done
 qed
 
 lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
   unfolding closure_def using islimpt_punctured by blast
 
 lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
   by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
 
 lemma bdd_below_closure:
   fixes A :: "real set"
   assumes "bdd_below A"
   shows "bdd_below (closure A)"
 proof -
   from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
     by (auto simp: bdd_below_def)
   then have "A \<subseteq> {m..}" by auto
   then have "closure A \<subseteq> {m..}"
     using closed_real_atLeast by (rule closure_minimal)
   then show ?thesis
     by (auto simp: bdd_below_def)
 qed
 
 
 subsection \<open>Frontier (also known as boundary)\<close>
 
 definition\<^marker>\<open>tag important\<close> frontier :: "('a::topological_space) set \<Rightarrow> 'a set" where
 "frontier S = closure S - interior S"
 
 lemma frontier_closed [iff]: "closed (frontier S)"
   by (simp add: frontier_def closed_Diff)
 
 lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
   by (auto simp: frontier_def interior_closure)
 
 lemma frontier_Int: "frontier(S \<inter> T) = closure(S \<inter> T) \<inter> (frontier S \<union> frontier T)"
 proof -
   have "closure (S \<inter> T) \<subseteq> closure S" "closure (S \<inter> T) \<subseteq> closure T"
     by (simp_all add: closure_mono)
   then show ?thesis
     by (auto simp: frontier_closures)
 qed
 
 lemma frontier_Int_subset: "frontier(S \<inter> T) \<subseteq> frontier S \<union> frontier T"
   by (auto simp: frontier_Int)
 
 lemma frontier_Int_closed:
   assumes "closed S" "closed T"
   shows "frontier(S \<inter> T) = (frontier S \<inter> T) \<union> (S \<inter> frontier T)"
 proof -
   have "closure (S \<inter> T) = T \<inter> S"
     using assms by (simp add: Int_commute closed_Int)
   moreover have "T \<inter> (closure S \<inter> closure (- S)) = frontier S \<inter> T"
     by (simp add: Int_commute frontier_closures)
   ultimately show ?thesis
     by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
 qed
 
 lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
   by (metis frontier_def closure_closed Diff_subset)
 
 lemma frontier_empty [simp]: "frontier {} = {}"
   by (simp add: frontier_def)
 
 lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
 proof -
   {
     assume "frontier S \<subseteq> S"
     then have "closure S \<subseteq> S"
       using interior_subset unfolding frontier_def by auto
     then have "closed S"
       using closure_subset_eq by auto
   }
   then show ?thesis using frontier_subset_closed[of S] ..
 qed
 
 lemma frontier_complement [simp]: "frontier (- S) = frontier S"
   by (auto simp: frontier_def closure_complement interior_complement)
 
 lemma frontier_Un_subset: "frontier(S \<union> T) \<subseteq> frontier S \<union> frontier T"
   by (metis compl_sup frontier_Int_subset frontier_complement)
 
 lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
   using frontier_complement frontier_subset_eq[of "- S"]
   unfolding open_closed by auto
 
 lemma frontier_UNIV [simp]: "frontier UNIV = {}"
   using frontier_complement frontier_empty by fastforce
 
 lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
   by (simp add: Int_commute frontier_def interior_closure)
 
 lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
   by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
 
 lemma closure_Un_frontier: "closure S = S \<union> frontier S"
 proof -
   have "S \<union> interior S = S"
     using interior_subset by auto
   then show ?thesis
     using closure_subset by (auto simp: frontier_def)
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Filters and the ``eventually true'' quantifier\<close>
 
 text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
 
 lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
 proof
   assume "trivial_limit (at a within S)"
   then show "\<not> a islimpt S"
     unfolding trivial_limit_def
     unfolding eventually_at_topological
     unfolding islimpt_def
     apply (clarsimp simp add: set_eq_iff)
     apply (rename_tac T, rule_tac x=T in exI)
     apply (clarsimp, drule_tac x=y in bspec, simp_all)
     done
 next
   assume "\<not> a islimpt S"
   then show "trivial_limit (at a within S)"
     unfolding trivial_limit_def eventually_at_topological islimpt_def
     by metis
 qed
 
 lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
   using trivial_limit_within [of a UNIV] by simp
 
 lemma trivial_limit_at: "\<not> trivial_limit (at a)"
   for a :: "'a::perfect_space"
   by (rule at_neq_bot)
 
 lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
   using islimpt_in_closure by (metis trivial_limit_within)
 
 lemma not_in_closure_trivial_limitI:
   "x \<notin> closure s \<Longrightarrow> trivial_limit (at x within s)"
   using not_trivial_limit_within[of x s]
   by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)
 
 lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
   if "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)"
   by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)
 
 lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
   using not_trivial_limit_within[of c A] by blast
 
 text \<open>Some property holds "sufficiently close" to the limit point.\<close>
 
 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
   by simp
 
 lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
   by (simp add: filter_eq_iff)
 
 lemma Lim_topological:
   "(f \<longlongrightarrow> l) net \<longleftrightarrow>
     trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
   unfolding tendsto_def trivial_limit_eq by auto
 
 lemma eventually_within_Un:
   "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
     eventually P (at x within s) \<and> eventually P (at x within t)"
   unfolding eventually_at_filter
   by (auto elim!: eventually_rev_mp)
 
 lemma Lim_within_union:
  "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
   (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
   unfolding tendsto_def
   by (auto simp: eventually_within_Un)
 
 
 subsection \<open>Limits\<close>
 
 text \<open>The expected monotonicity property.\<close>
 
 lemma Lim_Un:
   assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
   shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
   using assms unfolding at_within_union by (rule filterlim_sup)
 
 lemma Lim_Un_univ:
   "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
     S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
   by (metis Lim_Un)
 
 text \<open>Interrelations between restricted and unrestricted limits.\<close>
 
 lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
   by (metis order_refl filterlim_mono subset_UNIV at_le)
 
 lemma eventually_within_interior:
   assumes "x \<in> interior S"
   shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
   (is "?lhs = ?rhs")
 proof
   from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
   {
     assume ?lhs
     then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
       by (auto simp: eventually_at_topological)
     with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
       by auto
     then show ?rhs
       by (auto simp: eventually_at_topological)
   next
     assume ?rhs
     then show ?lhs
       by (auto elim: eventually_mono simp: eventually_at_filter)
   }
 qed
 
 lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
   unfolding filter_eq_iff by (intro allI eventually_within_interior)
 
 lemma Lim_within_LIMSEQ:
   fixes a :: "'a::first_countable_topology"
   assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at a within T)"
   using assms unfolding tendsto_def [where l=L]
   by (simp add: sequentially_imp_eventually_within)
 
 lemma Lim_right_bound:
   fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
     'b::{linorder_topology, conditionally_complete_linorder}"
   assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
     and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
   shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
 proof (cases "{x<..} \<inter> I = {}")
   case True
   then show ?thesis by simp
 next
   case False
   show ?thesis
   proof (rule order_tendstoI)
     fix a
     assume a: "a < Inf (f ` ({x<..} \<inter> I))"
     {
       fix y
       assume "y \<in> {x<..} \<inter> I"
       with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
         by (auto intro!: cInf_lower bdd_belowI2)
       with a have "a < f y"
         by (blast intro: less_le_trans)
     }
     then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
       by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
   next
     fix a
     assume "Inf (f ` ({x<..} \<inter> I)) < a"
     from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
       by auto
     then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
       unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
     then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
       unfolding eventually_at_filter by eventually_elim simp
   qed
 qed
 
 (*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
 lemma islimpt_sequential:
   fixes x :: "'a::first_countable_topology"
   shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
     (is "?lhs = ?rhs")
 proof
   assume ?lhs
   from countable_basis_at_decseq[of x] obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. x \<in> A i"
       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
   {
     fix n
     from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
       unfolding islimpt_def using A(1,2)[of n] by auto
     then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
       unfolding f_def by (rule someI_ex)
     then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
   }
   then have "\<forall>n. f n \<in> S - {x}" by auto
   moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "x \<in> S"
     from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
     show "eventually (\<lambda>x. f x \<in> S) sequentially"
       by (auto elim!: eventually_mono)
   qed
   ultimately show ?rhs by fast
 next
   assume ?rhs
   then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
     by auto
   show ?lhs
     unfolding islimpt_def
   proof safe
     fix T
     assume "open T" "x \<in> T"
     from lim[THEN topological_tendstoD, OF this] f
     show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
       unfolding eventually_sequentially by auto
   qed
 qed
 
 text\<open>These are special for limits out of the same topological space.\<close>
 
 lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
   unfolding id_def by (rule tendsto_ident_at)
 
 lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
   unfolding id_def by (rule tendsto_ident_at)
 
 text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
 
 abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
   where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
 
 lemma netlimit_at [simp]:
   fixes a :: "'a::{perfect_space,t2_space}"
   shows "netlimit (at a) = a"
   using Lim_ident_at [of a UNIV] by simp
 
 lemma lim_within_interior:
   "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
   by (metis at_within_interior)
 
 lemma netlimit_within_interior:
   fixes x :: "'a::{t2_space,perfect_space}"
   assumes "x \<in> interior S"
   shows "netlimit (at x within S) = x"
   using assms by (metis at_within_interior netlimit_at)
 
 text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
 
 lemma closure_sequential:
   fixes l :: "'a::first_countable_topology"
   shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
   (is "?lhs = ?rhs")
 proof
   assume "?lhs"
   moreover
   {
     assume "l \<in> S"
     then have "?rhs" using tendsto_const[of l sequentially] by auto
   }
   moreover
   {
     assume "l islimpt S"
     then have "?rhs" unfolding islimpt_sequential by auto
   }
   ultimately show "?rhs"
     unfolding closure_def by auto
 next
   assume "?rhs"
   then show "?lhs" unfolding closure_def islimpt_sequential by auto
 qed
 
 lemma closed_sequential_limits:
   fixes S :: "'a::first_countable_topology set"
   shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
 by (metis closure_sequential closure_subset_eq subset_iff)
 
 lemma tendsto_If_within_closures:
   assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
       (f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
   assumes g: "x \<in> t \<union> (closure s \<inter> closure t) \<Longrightarrow>
       (g \<longlongrightarrow> l x) (at x within t \<union> (closure s \<inter> closure t))"
   assumes "x \<in> s \<union> t"
   shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
 proof -
   have *: "(s \<union> t) \<inter> {x. x \<in> s} = s" "(s \<union> t) \<inter> {x. x \<notin> s} = t - s"
     by auto
   have "(f \<longlongrightarrow> l x) (at x within s)"
     by (rule filterlim_at_within_closure_implies_filterlim)
        (use \<open>x \<in> _\<close> in \<open>auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]\<close>)
   moreover
   have "(g \<longlongrightarrow> l x) (at x within t - s)"
     by (rule filterlim_at_within_closure_implies_filterlim)
       (use \<open>x \<in> _\<close> in
         \<open>auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset\<close>)
   ultimately show ?thesis
     by (intro filterlim_at_within_If) (simp_all only: *)
 qed
 
 
 subsection \<open>Compactness\<close>
 
 lemma brouwer_compactness_lemma:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
   assumes "compact s"
     and "continuous_on s f"
     and "\<not> (\<exists>x\<in>s. f x = 0)"
   obtains d where "0 < d" and "\<forall>x\<in>s. d \<le> norm (f x)"
 proof (cases "s = {}")
   case True
   show thesis
     by (rule that [of 1]) (auto simp: True)
 next
   case False
   have "continuous_on s (norm \<circ> f)"
     by (rule continuous_intros continuous_on_norm assms(2))+
   with False obtain x where x: "x \<in> s" "\<forall>y\<in>s. (norm \<circ> f) x \<le> (norm \<circ> f) y"
     using continuous_attains_inf[OF assms(1), of "norm \<circ> f"]
     unfolding o_def
     by auto
   have "(norm \<circ> f) x > 0"
     using assms(3) and x(1)
     by auto
   then show ?thesis
     by (rule that) (insert x(2), auto simp: o_def)
 qed
 
 subsubsection \<open>Bolzano-Weierstrass property\<close>
 
 proposition Heine_Borel_imp_Bolzano_Weierstrass:
   assumes "compact s"
     and "infinite t"
     and "t \<subseteq> s"
   shows "\<exists>x \<in> s. x islimpt t"
 proof (rule ccontr)
   assume "\<not> (\<exists>x \<in> s. x islimpt t)"
   then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
     unfolding islimpt_def
     using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
     by auto
   obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
     using assms(1)[unfolded compact_eq_Heine_Borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
     using f by auto
   from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
     by auto
   {
     fix x y
     assume "x \<in> t" "y \<in> t" "f x = f y"
     then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"
       using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
     then have "x = y"
       using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
       by auto
   }
   then have "inj_on f t"
     unfolding inj_on_def by simp
   then have "infinite (f ` t)"
     using assms(2) using finite_imageD by auto
   moreover
   {
     fix x
     assume "x \<in> t" "f x \<notin> g"
     from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
       by auto
     then obtain y where "y \<in> s" "h = f y"
       using g'[THEN bspec[where x=h]] by auto
     then have "y = x"
       using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
       by auto
     then have False
       using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
       by auto
   }
   then have "f ` t \<subseteq> g" by auto
   ultimately show False
     using g(2) using finite_subset by auto
 qed
 
 lemma sequence_infinite_lemma:
   fixes f :: "nat \<Rightarrow> 'a::t1_space"
   assumes "\<forall>n. f n \<noteq> l"
     and "(f \<longlongrightarrow> l) sequentially"
   shows "infinite (range f)"
 proof
   assume "finite (range f)"
   then have "closed (range f)"
     by (rule finite_imp_closed)
   then have "open (- range f)"
     by (rule open_Compl)
   from assms(1) have "l \<in> - range f"
     by auto
   from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
     using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
     by (rule topological_tendstoD)
   then show False
     unfolding eventually_sequentially
     by auto
 qed
 
 lemma Bolzano_Weierstrass_imp_closed:
   fixes s :: "'a::{first_countable_topology,t2_space} set"
   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
   shows "closed s"
 proof -
   {
     fix x l
     assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
     then have "l \<in> s"
     proof (cases "\<forall>n. x n \<noteq> l")
       case False
       then show "l\<in>s" using as(1) by auto
     next
       case True note cas = this
       with as(2) have "infinite (range x)"
         using sequence_infinite_lemma[of x l] by auto
       then obtain l' where "l'\<in>s" "l' islimpt (range x)"
         using assms[THEN spec[where x="range x"]] as(1) by auto
       then show "l\<in>s" using sequence_unique_limpt[of x l l']
         using as cas by auto
     qed
   }
   then show ?thesis
     unfolding closed_sequential_limits by fast
 qed
 
 lemma closure_insert:
   fixes x :: "'a::t1_space"
   shows "closure (insert x s) = insert x (closure s)"
   apply (rule closure_unique)
   apply (rule insert_mono [OF closure_subset])
   apply (rule closed_insert [OF closed_closure])
   apply (simp add: closure_minimal)
   done
 
 
 text\<open>In particular, some common special cases.\<close>
 
 lemma compact_Un [intro]:
   assumes "compact s"
     and "compact t"
   shows " compact (s \<union> t)"
 proof (rule compactI)
   fix f
   assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
   from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
   moreover
   from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
     unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
   ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
     by (auto intro!: exI[of _ "s' \<union> t'"])
 qed
 
 lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
   by (induct set: finite) auto
 
 lemma compact_UN [intro]:
   "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
   by (rule compact_Union) auto
 
 lemma closed_Int_compact [intro]:
   assumes "closed s"
     and "compact t"
   shows "compact (s \<inter> t)"
   using compact_Int_closed [of t s] assms
   by (simp add: Int_commute)
 
 lemma compact_Int [intro]:
   fixes s t :: "'a :: t2_space set"
   assumes "compact s"
     and "compact t"
   shows "compact (s \<inter> t)"
   using assms by (intro compact_Int_closed compact_imp_closed)
 
 lemma compact_sing [simp]: "compact {a}"
   unfolding compact_eq_Heine_Borel by auto
 
 lemma compact_insert [simp]:
   assumes "compact s"
   shows "compact (insert x s)"
 proof -
   have "compact ({x} \<union> s)"
     using compact_sing assms by (rule compact_Un)
   then show ?thesis by simp
 qed
 
 lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
   by (induct set: finite) simp_all
 
 lemma open_delete:
   fixes s :: "'a::t1_space set"
   shows "open s \<Longrightarrow> open (s - {x})"
   by (simp add: open_Diff)
 
 
 text\<open>Compactness expressed with filters\<close>
 
 lemma closure_iff_nhds_not_empty:
   "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
 proof safe
   assume x: "x \<in> closure X"
   fix S A
   assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
   then have "x \<notin> closure (-S)"
     by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
   with x have "x \<in> closure X - closure (-S)"
     by auto
   also have "\<dots> \<subseteq> closure (X \<inter> S)"
     using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
   finally have "X \<inter> S \<noteq> {}" by auto
   then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
 next
   assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
   from this[THEN spec, of "- X", THEN spec, of "- closure X"]
   show "x \<in> closure X"
     by (simp add: closure_subset open_Compl)
 qed
 
 lemma compact_filter:
   "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
 proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
   fix F
   assume "compact U"
   assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
   then have "U \<noteq> {}"
     by (auto simp: eventually_False)
 
   define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
   then have "\<forall>z\<in>Z. closed z"
     by auto
   moreover
   have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
     unfolding Z_def by (auto elim: eventually_mono intro: subsetD[OF closure_subset])
   have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
   proof (intro allI impI)
     fix B assume "finite B" "B \<subseteq> Z"
     with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
       by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
     with F show "U \<inter> \<Inter>B \<noteq> {}"
       by (intro notI) (simp add: eventually_False)
   qed
   ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
     using \<open>compact U\<close> unfolding compact_fip by blast
   then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
     by auto
 
   have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
     unfolding eventually_inf eventually_nhds
   proof safe
     fix P Q R S
     assume "eventually R F" "open S" "x \<in> S"
     with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
     have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
     moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
     ultimately show False by (auto simp: set_eq_iff)
   qed
   with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
     by (metis eventually_bot)
 next
   fix A
   assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
   define F where "F = (INF a\<in>insert U A. principal a)"
   have "F \<noteq> bot"
     unfolding F_def
   proof (rule INF_filter_not_bot)
     fix X
     assume X: "X \<subseteq> insert U A" "finite X"
     with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
       by auto
     with X show "(INF a\<in>X. principal a) \<noteq> bot"
       by (auto simp: INF_principal_finite principal_eq_bot_iff)
   qed
   moreover
   have "F \<le> principal U"
     unfolding F_def by auto
   then have "eventually (\<lambda>x. x \<in> U) F"
     by (auto simp: le_filter_def eventually_principal)
   moreover
   assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
   ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
     by auto
 
   { fix V assume "V \<in> A"
     then have "F \<le> principal V"
       unfolding F_def by (intro INF_lower2[of V]) auto
     then have V: "eventually (\<lambda>x. x \<in> V) F"
       by (auto simp: le_filter_def eventually_principal)
     have "x \<in> closure V"
       unfolding closure_iff_nhds_not_empty
     proof (intro impI allI)
       fix S A
       assume "open S" "x \<in> S" "S \<subseteq> A"
       then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
         by (auto simp: eventually_nhds)
       with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
         by (auto simp: eventually_inf)
       with x show "V \<inter> A \<noteq> {}"
         by (auto simp del: Int_iff simp add: trivial_limit_def)
     qed
     then have "x \<in> V"
       using \<open>V \<in> A\<close> A(1) by simp
   }
   with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
   with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
 qed
 
 definition\<^marker>\<open>tag important\<close> countably_compact :: "('a::topological_space) set \<Rightarrow> bool" where
 "countably_compact U \<longleftrightarrow>
   (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A
      \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
 
 lemma countably_compactE:
   assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
   obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
   using assms unfolding countably_compact_def by metis
 
 lemma countably_compactI:
   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
   shows "countably_compact s"
   using assms unfolding countably_compact_def by metis
 
 lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
   by (auto simp: compact_eq_Heine_Borel countably_compact_def)
 
 lemma countably_compact_imp_compact:
   assumes "countably_compact U"
     and ccover: "countable B" "\<forall>b\<in>B. open b"
     and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
   shows "compact U"
   using \<open>countably_compact U\<close>
   unfolding compact_eq_Heine_Borel countably_compact_def
 proof safe
   fix A
   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
   assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
   moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
   ultimately have "countable C" "\<forall>a\<in>C. open a"
     unfolding C_def using ccover by auto
   moreover
   have "\<Union>A \<inter> U \<subseteq> \<Union>C"
   proof safe
     fix x a
     assume "x \<in> U" "x \<in> a" "a \<in> A"
     with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
       by blast
     with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
       unfolding C_def by auto
   qed
   then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
   ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
     using * by metis
   then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
     by (auto simp: C_def)
   then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
     unfolding bchoice_iff Bex_def ..
   with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
     unfolding C_def by (intro exI[of _ "f`T"]) fastforce
 qed
 
 proposition countably_compact_imp_compact_second_countable:
   "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
 proof (rule countably_compact_imp_compact)
   fix T and x :: 'a
   assume "open T" "x \<in> T"
   from topological_basisE[OF is_basis this] obtain b where
     "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
   then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
     by blast
 qed (insert countable_basis topological_basis_open[OF is_basis], auto)
 
 lemma countably_compact_eq_compact:
   "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
   using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
 
 subsubsection\<open>Sequential compactness\<close>
 
 definition\<^marker>\<open>tag important\<close> seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
 "seq_compact S \<longleftrightarrow>
   (\<forall>f. (\<forall>n. f n \<in> S)
     \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
 
 lemma seq_compactI:
   assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
   shows "seq_compact S"
   unfolding seq_compact_def using assms by fast
 
 lemma seq_compactE:
   assumes "seq_compact S" "\<forall>n. f n \<in> S"
   obtains l r where "l \<in> S" "strict_mono (r :: nat \<Rightarrow> nat)" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
   using assms unfolding seq_compact_def by fast
 
 lemma closed_sequentially: (* TODO: move upwards *)
   assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
   shows "l \<in> s"
 proof (rule ccontr)
   assume "l \<notin> s"
   with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
     by (fast intro: topological_tendstoD)
   with \<open>\<forall>n. f n \<in> s\<close> show "False"
     by simp
 qed
 
 lemma seq_compact_Int_closed:
   assumes "seq_compact s" and "closed t"
   shows "seq_compact (s \<inter> t)"
 proof (rule seq_compactI)
   fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
   hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
     by simp_all
   from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
   obtain l r where "l \<in> s" and r: "strict_mono r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
     by (rule seq_compactE)
   from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
     by simp
   from \<open>closed t\<close> and this and l have "l \<in> t"
     by (rule closed_sequentially)
   with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
     by fast
 qed
 
 lemma seq_compact_closed_subset:
   assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
   shows "seq_compact s"
   using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
 
 lemma seq_compact_imp_countably_compact:
   fixes U :: "'a :: first_countable_topology set"
   assumes "seq_compact U"
   shows "countably_compact U"
 proof (safe intro!: countably_compactI)
   fix A
   assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
   have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> strict_mono (r :: nat \<Rightarrow> nat) \<and> (X \<circ> r) \<longlonglongrightarrow> x"
     using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
   show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
   proof cases
     assume "finite A"
     with A show ?thesis by auto
   next
     assume "infinite A"
     then have "A \<noteq> {}" by auto
     show ?thesis
     proof (rule ccontr)
       assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
       then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
         by auto
       then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
         by metis
       define X where "X n = X' (from_nat_into A ` {.. n})" for n
       have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
         using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
       then have "range X \<subseteq> U"
         by auto
       with subseq[of X] obtain r x where "x \<in> U" and r: "strict_mono r" "(X \<circ> r) \<longlonglongrightarrow> x"
         by auto
       from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
       obtain n where "x \<in> from_nat_into A n" by auto
       with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
       have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
         unfolding tendsto_def by (auto simp: comp_def)
       then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
         by (auto simp: eventually_sequentially)
       moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
         by auto
       moreover from \<open>strict_mono r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
         by (auto intro!: exI[of _ "max n N"])
       ultimately show False
         by auto
     qed
   qed
 qed
 
 lemma compact_imp_seq_compact:
   fixes U :: "'a :: first_countable_topology set"
   assumes "compact U"
   shows "seq_compact U"
   unfolding seq_compact_def
 proof safe
   fix X :: "nat \<Rightarrow> 'a"
   assume "\<forall>n. X n \<in> U"
   then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
     by (auto simp: eventually_filtermap)
   moreover
   have "filtermap X sequentially \<noteq> bot"
     by (simp add: trivial_limit_def eventually_filtermap)
   ultimately
   obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
     using \<open>compact U\<close> by (auto simp: compact_filter)
 
   from countable_basis_at_decseq[of x]
   obtain A where A:
       "\<And>i. open (A i)"
       "\<And>i. x \<in> A i"
       "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by blast
   define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
   {
     fix n i
     have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
     proof (rule ccontr)
       assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
       then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
         by auto
       then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
         by (auto simp: eventually_filtermap eventually_sequentially)
       moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
         using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
       ultimately have "eventually (\<lambda>x. False) ?F"
         by (auto simp: eventually_inf)
       with x show False
         by (simp add: eventually_False)
     qed
     then have "i < s n i" "X (s n i) \<in> A (Suc n)"
       unfolding s_def by (auto intro: someI2_ex)
   }
   note s = this
   define r where "r = rec_nat (s 0 0) s"
   have "strict_mono r"
     by (auto simp: r_def s strict_mono_Suc_iff)
   moreover
   have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
   proof (rule topological_tendstoI)
     fix S
     assume "open S" "x \<in> S"
     with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
       by auto
     moreover
     {
       fix i
       assume "Suc 0 \<le> i"
       then have "X (r i) \<in> A i"
         by (cases i) (simp_all add: r_def s)
     }
     then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
       by (auto simp: eventually_sequentially)
     ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
       by eventually_elim auto
   qed
   ultimately show "\<exists>x \<in> U. \<exists>r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
     using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
 qed
 
 lemma countably_compact_imp_acc_point:
   assumes "countably_compact s"
     and "countable t"
     and "infinite t"
     and "t \<subseteq> s"
   shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
 proof (rule ccontr)
   define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
   note \<open>countably_compact s\<close>
   moreover have "\<forall>t\<in>C. open t"
     by (auto simp: C_def)
   moreover
   assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
   then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
   have "s \<subseteq> \<Union>C"
     using \<open>t \<subseteq> s\<close>
     unfolding C_def
     apply (safe dest!: s)
     apply (rule_tac a="U \<inter> t" in UN_I)
     apply (auto intro!: interiorI simp add: finite_subset)
     done
   moreover
   from \<open>countable t\<close> have "countable C"
     unfolding C_def by (auto intro: countable_Collect_finite_subset)
   ultimately
   obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
     by (rule countably_compactE)
   then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
     and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
     by (metis (lifting) finite_subset_image C_def)
   from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
     using interior_subset by blast
   moreover have "finite (\<Union>E)"
     using E by auto
   ultimately show False using \<open>infinite t\<close>
     by (auto simp: finite_subset)
 qed
 
 lemma countable_acc_point_imp_seq_compact:
   fixes s :: "'a::first_countable_topology set"
   assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
     (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
   shows "seq_compact s"
 proof -
   {
     fix f :: "nat \<Rightarrow> 'a"
     assume f: "\<forall>n. f n \<in> s"
     have "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
     proof (cases "finite (range f)")
       case True
       obtain l where "infinite {n. f n = f l}"
         using pigeonhole_infinite[OF _ True] by auto
       then obtain r :: "nat \<Rightarrow> nat" where "strict_mono  r" and fr: "\<forall>n. f (r n) = f l"
         using infinite_enumerate by blast
       then have "strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
         by (simp add: fr o_def)
       with f show "\<exists>l\<in>s. \<exists>r. strict_mono  r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
         by auto
     next
       case False
       with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
         by auto
       then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
       from this(2) have "\<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
         using acc_point_range_imp_convergent_subsequence[of l f] by auto
       with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
     qed
   }
   then show ?thesis
     unfolding seq_compact_def by auto
 qed
 
 lemma seq_compact_eq_countably_compact:
   fixes U :: "'a :: first_countable_topology set"
   shows "seq_compact U \<longleftrightarrow> countably_compact U"
   using
     countable_acc_point_imp_seq_compact
     countably_compact_imp_acc_point
     seq_compact_imp_countably_compact
   by metis
 
 lemma seq_compact_eq_acc_point:
   fixes s :: "'a :: first_countable_topology set"
   shows "seq_compact s \<longleftrightarrow>
     (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
   using
     countable_acc_point_imp_seq_compact[of s]
     countably_compact_imp_acc_point[of s]
     seq_compact_imp_countably_compact[of s]
   by metis
 
 lemma seq_compact_eq_compact:
   fixes U :: "'a :: second_countable_topology set"
   shows "seq_compact U \<longleftrightarrow> compact U"
   using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
 
 proposition Bolzano_Weierstrass_imp_seq_compact:
   fixes s :: "'a::{t1_space, first_countable_topology} set"
   shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
   by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Cartesian products\<close>
 
 lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
   unfolding seq_compact_def
   apply clarify
   apply (drule_tac x="fst \<circ> f" in spec)
   apply (drule mp, simp add: mem_Times_iff)
   apply (clarify, rename_tac l1 r1)
   apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
   apply (drule mp, simp add: mem_Times_iff)
   apply (clarify, rename_tac l2 r2)
   apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
   apply (rule_tac x="r1 \<circ> r2" in exI)
   apply (rule conjI, simp add: strict_mono_def)
   apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
   apply (drule (1) tendsto_Pair) back
   apply (simp add: o_def)
   done
 
 lemma compact_Times:
   assumes "compact s" "compact t"
   shows "compact (s \<times> t)"
 proof (rule compactI)
   fix C
   assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
   have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
   proof
     fix x
     assume "x \<in> s"
     have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
     proof
       fix y
       assume "y \<in> t"
       with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
       then show "?P y" by (auto elim!: open_prod_elim)
     qed
     then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
       and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
       by metis
     then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
     with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
       by metis
     moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
       by (fastforce simp: subset_eq)
     ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
       using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
   qed
   then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
     and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>(d x)"
     unfolding subset_eq UN_iff by metis
   moreover
   from compactE_image[OF \<open>compact s\<close> a]
   obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
     by auto
   moreover
   {
     from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
       by auto
     also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>(d x))"
       using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
     finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>(d x))" .
   }
   ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
     by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
 qed
 
 
 lemma tube_lemma:
   assumes "compact K"
   assumes "open W"
   assumes "{x0} \<times> K \<subseteq> W"
   shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
 proof -
   {
     fix y assume "y \<in> K"
     then have "(x0, y) \<in> W" using assms by auto
     with \<open>open W\<close>
     have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
       by (rule open_prod_elim) blast
   }
   then obtain X0 Y where
     *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
     by metis
   from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
   with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
     by (meson compactE)
   then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
     by (force intro!: choice)
   with * CC show ?thesis
     by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
 qed
 
 lemma continuous_on_prod_compactE:
   fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
     and e::real
   assumes cont_fx: "continuous_on (U \<times> C) fx"
   assumes "compact C"
   assumes [intro]: "x0 \<in> U"
   notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
   assumes "e > 0"
   obtains X0 where "x0 \<in> X0" "open X0"
     "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
 proof -
   define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
   define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
   have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
     by (auto simp: vimage_def W0_def)
   have "open {..<e}" by simp
   have "continuous_on (U \<times> C) psi"
     by (auto intro!: continuous_intros simp: psi_def split_beta')
   from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
   obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
     unfolding W0_eq by blast
   have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
     unfolding W
     by (auto simp: W0_def psi_def \<open>0 < e\<close>)
   then have "{x0} \<times> C \<subseteq> W" by blast
   from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
   obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
     by blast
 
   have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
   proof safe
     fix x assume x: "x \<in> X0" "x \<in> U"
     fix t assume t: "t \<in> C"
     have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
       by (auto simp: psi_def)
     also
     {
       have "(x, t) \<in> X0 \<times> C"
         using t x
         by auto
       also note \<open>\<dots> \<subseteq> W\<close>
       finally have "(x, t) \<in> W" .
       with t x have "(x, t) \<in> W \<inter> U \<times> C"
         by blast
       also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
       finally  have "psi (x, t) < e"
         by (auto simp: W0_def)
     }
     finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
   qed
   from X0(1,2) this show ?thesis ..
 qed
 
 
 subsection \<open>Continuity\<close>
 
 lemma continuous_at_imp_continuous_within:
   "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
   unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
 
 lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
   by simp
 
 lemmas continuous_on = continuous_on_def \<comment> \<open>legacy theorem name\<close>
 
 lemma continuous_within_subset:
   "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
   unfolding continuous_within by(metis tendsto_within_subset)
 
 lemma continuous_on_interior:
   "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
   by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
 
 lemma continuous_on_eq:
   "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
   unfolding continuous_on_def tendsto_def eventually_at_topological
   by simp
 
 text \<open>Characterization of various kinds of continuity in terms of sequences.\<close>
 
 lemma continuous_within_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u::nat \<Rightarrow> 'a. u \<longlonglongrightarrow> a \<Longrightarrow> (\<forall>n. u n \<in> s) \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous (at a within s) f"
   using assms unfolding continuous_within tendsto_def[where l = "f a"]
   by (auto intro!: sequentially_imp_eventually_within)
 
 lemma continuous_within_tendsto_compose:
   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
   assumes "continuous (at a within s) f"
           "eventually (\<lambda>n. x n \<in> s) F"
           "(x \<longlongrightarrow> a) F "
   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
 proof -
   have *: "filterlim x (inf (nhds a) (principal s)) F"
     using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
   show ?thesis
     by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
 qed
 
 lemma continuous_within_tendsto_compose':
   fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
   assumes "continuous (at a within s) f"
     "\<And>n. x n \<in> s"
     "(x \<longlongrightarrow> a) F "
   shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
   by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
 
 lemma continuous_within_sequentially:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   shows "continuous (at a within s) f \<longleftrightarrow>
     (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
          \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
   using continuous_within_tendsto_compose'[of a s f _ sequentially]
     continuous_within_sequentiallyI[of a s f]
   by (auto simp: o_def)
 
 lemma continuous_at_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u. u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous (at a) f"
   using continuous_within_sequentiallyI[of a UNIV f] assms by auto
 
 lemma continuous_at_sequentially:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
   shows "continuous (at a) f \<longleftrightarrow>
     (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
   using continuous_within_sequentially[of a UNIV f] by simp
 
 lemma continuous_on_sequentiallyI:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   assumes "\<And>u a. (\<forall>n. u n \<in> s) \<Longrightarrow> a \<in> s \<Longrightarrow> u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
   shows "continuous_on s f"
   using assms unfolding continuous_on_eq_continuous_within
   using continuous_within_sequentiallyI[of _ s f] by auto
 
 lemma continuous_on_sequentially:
   fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
   shows "continuous_on s f \<longleftrightarrow>
     (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
       --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
     (is "?lhs = ?rhs")
 proof
   assume ?rhs
   then show ?lhs
     using continuous_within_sequentially[of _ s f]
     unfolding continuous_on_eq_continuous_within
     by auto
 next
   assume ?lhs
   then show ?rhs
     unfolding continuous_on_eq_continuous_within
     using continuous_within_sequentially[of _ s f]
     by auto
 qed
 
 text \<open>Continuity in terms of open preimages.\<close>
 
 lemma continuous_at_open:
   "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
   unfolding continuous_within_topological [of x UNIV f]
   unfolding imp_conjL
   by (intro all_cong imp_cong ex_cong conj_cong refl) auto
 
 lemma continuous_imp_tendsto:
   assumes "continuous (at x0) f"
     and "x \<longlonglongrightarrow> x0"
   shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
 proof (rule topological_tendstoI)
   fix S
   assume "open S" "f x0 \<in> S"
   then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
      using assms continuous_at_open by metis
   then have "eventually (\<lambda>n. x n \<in> T) sequentially"
     using assms T_def by (auto simp: tendsto_def)
   then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
     using T_def by (auto elim!: eventually_mono)
 qed
 
 subsection \<open>Homeomorphisms\<close>
 
 definition\<^marker>\<open>tag important\<close> "homeomorphism s t f g \<longleftrightarrow>
   (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
   (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
 
 lemma homeomorphismI [intro?]:
   assumes "continuous_on S f" "continuous_on T g"
           "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
     shows "homeomorphism S T f g"
   using assms by (force simp: homeomorphism_def)
 
 lemma homeomorphism_translation:
   fixes a :: "'a :: real_normed_vector"
   shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
 unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
 
 lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
   by (rule homeomorphismI) auto
 
 lemma homeomorphism_compose:
   assumes "homeomorphism S T f g" "homeomorphism T U h k"
     shows "homeomorphism S U (h o f) (g o k)"
   using assms
   unfolding homeomorphism_def
   by (intro conjI ballI continuous_on_compose) (auto simp: image_iff)
 
 lemma homeomorphism_cong:
   "homeomorphism X' Y' f' g'"
     if "homeomorphism X Y f g" "X' = X" "Y' = Y" "\<And>x. x \<in> X \<Longrightarrow> f' x = f x" "\<And>y. y \<in> Y \<Longrightarrow> g' y = g y"
   using that by (auto simp add: homeomorphism_def)
 
 lemma homeomorphism_empty [simp]:
   "homeomorphism {} {} f g"
   unfolding homeomorphism_def by auto
 
 lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
   by (force simp: homeomorphism_def)
 
 definition\<^marker>\<open>tag important\<close> homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
     (infixr "homeomorphic" 60)
   where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
 
 lemma homeomorphic_empty [iff]:
      "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
   by (auto simp: homeomorphic_def homeomorphism_def)
 
 lemma homeomorphic_refl: "s homeomorphic s"
   unfolding homeomorphic_def homeomorphism_def
   using continuous_on_id
   apply (rule_tac x = "(\<lambda>x. x)" in exI)
   apply (rule_tac x = "(\<lambda>x. x)" in exI)
   apply blast
   done
 
 lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
   unfolding homeomorphic_def homeomorphism_def
   by blast
 
 lemma homeomorphic_trans [trans]:
   assumes "S homeomorphic T"
       and "T homeomorphic U"
     shows "S homeomorphic U"
   using assms
   unfolding homeomorphic_def
 by (metis homeomorphism_compose)
 
 lemma homeomorphic_minimal:
   "s homeomorphic t \<longleftrightarrow>
     (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
            (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
            continuous_on s f \<and> continuous_on t g)"
    (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then show ?rhs
     by (fastforce simp: homeomorphic_def homeomorphism_def)
 next
   assume ?rhs
   then show ?lhs
     apply clarify
     unfolding homeomorphic_def homeomorphism_def
     by (metis equalityI image_subset_iff subsetI)
  qed
 
 lemma homeomorphicI [intro?]:
    "\<lbrakk>f ` S = T; g ` T = S;
      continuous_on S f; continuous_on T g;
      \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
      \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
 unfolding homeomorphic_def homeomorphism_def by metis
 
 lemma homeomorphism_of_subsets:
    "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
     \<Longrightarrow> homeomorphism S' T' f g"
 apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
 by (metis subsetD imageI)
 
 lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
   by (simp add: homeomorphism_def)
 
 lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
   by (simp add: homeomorphism_def)
 
 lemma continuous_on_no_limpt:
    "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
   unfolding continuous_on_def
   by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
 
 lemma continuous_on_finite:
   fixes S :: "'a::t1_space set"
   shows "finite S \<Longrightarrow> continuous_on S f"
 by (metis continuous_on_no_limpt islimpt_finite)
 
 lemma homeomorphic_finite:
   fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
   assumes "finite T"
   shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
 proof
   assume "S homeomorphic T"
   with assms show ?rhs
     apply (auto simp: homeomorphic_def homeomorphism_def)
      apply (metis finite_imageI)
     by (metis card_image_le finite_imageI le_antisym)
 next
   assume R: ?rhs
   with finite_same_card_bij obtain h where "bij_betw h S T"
     by auto
   with R show ?lhs
     apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
     apply (rule_tac x=h in exI)
     apply (rule_tac x="inv_into S h" in exI)
     apply (auto simp:  bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
     apply (metis bij_betw_def bij_betw_inv_into)
     done
 qed
 
 text \<open>Relatively weak hypotheses if a set is compact.\<close>
 
 lemma homeomorphism_compact:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes "compact s" "continuous_on s f"  "f ` s = t"  "inj_on f s"
   shows "\<exists>g. homeomorphism s t f g"
 proof -
   define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
   have g: "\<forall>x\<in>s. g (f x) = x"
     using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
   {
     fix y
     assume "y \<in> t"
     then obtain x where x:"f x = y" "x\<in>s"
       using assms(3) by auto
     then have "g (f x) = x" using g by auto
     then have "f (g y) = y" unfolding x(1)[symmetric] by auto
   }
   then have g':"\<forall>x\<in>t. f (g x) = x" by auto
   moreover
   {
     fix x
     have "x\<in>s \<Longrightarrow> x \<in> g ` t"
       using g[THEN bspec[where x=x]]
       unfolding image_iff
       using assms(3)
       by (auto intro!: bexI[where x="f x"])
     moreover
     {
       assume "x\<in>g ` t"
       then obtain y where y:"y\<in>t" "g y = x" by auto
       then obtain x' where x':"x'\<in>s" "f x' = y"
         using assms(3) by auto
       then have "x \<in> s"
         unfolding g_def
         using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
         unfolding y(2)[symmetric] and g_def
         by auto
     }
     ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
   }
   then have "g ` t = s" by auto
   ultimately show ?thesis
     unfolding homeomorphism_def homeomorphic_def
     apply (rule_tac x=g in exI)
     using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
     apply auto
     done
 qed
 
 lemma homeomorphic_compact:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
   unfolding homeomorphic_def by (metis homeomorphism_compact)
 
 text\<open>Preservation of topological properties.\<close>
 
 lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
   unfolding homeomorphic_def homeomorphism_def
   by (metis compact_continuous_image)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>On Linorder Topologies\<close>
 
 lemma islimpt_greaterThanLessThan1:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows  "a islimpt {a<..<b}"
 proof (rule islimptI)
   fix T
   assume "open T" "a \<in> T"
   from open_right[OF this \<open>a < b\<close>]
   obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
   with assms dense[of a "min c b"]
   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
     by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
       not_le order.strict_implies_order subset_eq)
 qed
 
 lemma islimpt_greaterThanLessThan2:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows  "b islimpt {a<..<b}"
 proof (rule islimptI)
   fix T
   assume "open T" "b \<in> T"
   from open_left[OF this \<open>a < b\<close>]
   obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
   with assms dense[of "max a c" b]
   show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
     by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
       not_le order.strict_implies_order subset_eq)
 qed
 
 lemma closure_greaterThanLessThan[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
 proof
   have "?l \<subseteq> closure ?r"
     by (rule closure_mono) auto
   thus "closure {a<..<b} \<subseteq> {a..b}" by simp
 qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
   islimpt_greaterThanLessThan2)
 
 lemma closure_greaterThan[simp]:
   fixes a b::"'a::{no_top, linorder_topology, dense_order}"
   shows "closure {a<..} = {a..}"
 proof -
   from gt_ex obtain b where "a < b" by auto
   hence "{a<..} = {a<..<b} \<union> {b..}" by auto
   also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
     by auto
   finally show ?thesis .
 qed
 
 lemma closure_lessThan[simp]:
   fixes b::"'a::{no_bot, linorder_topology, dense_order}"
   shows "closure {..<b} = {..b}"
 proof -
   from lt_ex obtain a where "a < b" by auto
   hence "{..<b} = {a<..<b} \<union> {..a}" by auto
   also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
     by auto
   finally show ?thesis .
 qed
 
 lemma closure_atLeastLessThan[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows "closure {a ..< b} = {a .. b}"
 proof -
   from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
   also have "closure \<dots> = {a .. b}" unfolding closure_Un
     by (auto simp: assms less_imp_le)
   finally show ?thesis .
 qed
 
 lemma closure_greaterThanAtMost[simp]:
   fixes a b::"'a::{linorder_topology, dense_order}"
   assumes "a < b"
   shows "closure {a <.. b} = {a .. b}"
 proof -
   from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
   also have "closure \<dots> = {a .. b}" unfolding closure_Un
     by (auto simp: assms less_imp_le)
   finally show ?thesis .
 qed
 
 end
\ No newline at end of file
diff --git a/src/HOL/Topological_Spaces.thy b/src/HOL/Topological_Spaces.thy
--- a/src/HOL/Topological_Spaces.thy
+++ b/src/HOL/Topological_Spaces.thy
@@ -1,3824 +1,3827 @@
 (*  Title:      HOL/Topological_Spaces.thy
     Author:     Brian Huffman
     Author:     Johannes Hölzl
 *)
 
 section \<open>Topological Spaces\<close>
 
 theory Topological_Spaces
   imports Main
 begin
 
 named_theorems continuous_intros "structural introduction rules for continuity"
 
 subsection \<open>Topological space\<close>
 
 class "open" =
   fixes "open" :: "'a set \<Rightarrow> bool"
 
 class topological_space = "open" +
   assumes open_UNIV [simp, intro]: "open UNIV"
   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
 begin
 
 definition closed :: "'a set \<Rightarrow> bool"
   where "closed S \<longleftrightarrow> open (- S)"
 
 lemma open_empty [continuous_intros, intro, simp]: "open {}"
   using open_Union [of "{}"] by simp
 
 lemma open_Un [continuous_intros, intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
   using open_Union [of "{S, T}"] by simp
 
 lemma open_UN [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
   using open_Union [of "B ` A"] by simp
 
 lemma open_Inter [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
   by (induct set: finite) auto
 
 lemma open_INT [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
   using open_Inter [of "B ` A"] by simp
 
 lemma openI:
   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
   shows "open S"
 proof -
   have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
   moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
   ultimately show "open S" by simp
 qed
 
+lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
+by (auto intro: openI)
+
 lemma closed_empty [continuous_intros, intro, simp]: "closed {}"
   unfolding closed_def by simp
 
 lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
   unfolding closed_def by auto
 
 lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV"
   unfolding closed_def by simp
 
 lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
   unfolding closed_def by auto
 
 lemma closed_INT [continuous_intros, intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
   unfolding closed_def by auto
 
 lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter>K)"
   unfolding closed_def uminus_Inf by auto
 
 lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
   by (induct set: finite) auto
 
 lemma closed_UN [continuous_intros, intro]:
   "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
   using closed_Union [of "B ` A"] by simp
 
 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
   by (simp add: closed_def)
 
 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
   by (rule closed_def)
 
 lemma open_Diff [continuous_intros, intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
   by (simp add: closed_open Diff_eq open_Int)
 
 lemma closed_Diff [continuous_intros, intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
   by (simp add: open_closed Diff_eq closed_Int)
 
 lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)"
   by (simp add: closed_open)
 
 lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)"
   by (simp add: open_closed)
 
 lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
   unfolding Collect_neg_eq by (rule open_Compl)
 
 lemma open_Collect_conj:
   assumes "open {x. P x}" "open {x. Q x}"
   shows "open {x. P x \<and> Q x}"
   using open_Int[OF assms] by (simp add: Int_def)
 
 lemma open_Collect_disj:
   assumes "open {x. P x}" "open {x. Q x}"
   shows "open {x. P x \<or> Q x}"
   using open_Un[OF assms] by (simp add: Un_def)
 
 lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}"
   using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp
 
 lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}"
   unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg)
 
 lemma open_Collect_const: "open {x. P}"
   by (cases P) auto
 
 lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
   unfolding Collect_neg_eq by (rule closed_Compl)
 
 lemma closed_Collect_conj:
   assumes "closed {x. P x}" "closed {x. Q x}"
   shows "closed {x. P x \<and> Q x}"
   using closed_Int[OF assms] by (simp add: Int_def)
 
 lemma closed_Collect_disj:
   assumes "closed {x. P x}" "closed {x. Q x}"
   shows "closed {x. P x \<or> Q x}"
   using closed_Un[OF assms] by (simp add: Un_def)
 
 lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}"
   using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] by (simp add: Collect_all_eq)
 
 lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}"
   unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)
 
 lemma closed_Collect_const: "closed {x. P}"
   by (cases P) auto
 
 end
 
 
 subsection \<open>Hausdorff and other separation properties\<close>
 
 class t0_space = topological_space +
   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
 
 class t1_space = topological_space +
   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
 
 instance t1_space \<subseteq> t0_space
   by standard (fast dest: t1_space)
 
 context t1_space begin
 
 lemma separation_t1: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
   using t1_space[of x y] by blast
 
 lemma closed_singleton [iff]: "closed {a}"
 proof -
   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
   have "open ?T"
     by (simp add: open_Union)
   also have "?T = - {a}"
     by (auto simp add: set_eq_iff separation_t1)
   finally show "closed {a}"
     by (simp only: closed_def)
 qed
 
 lemma closed_insert [continuous_intros, simp]:
   assumes "closed S"
   shows "closed (insert a S)"
 proof -
   from closed_singleton assms have "closed ({a} \<union> S)"
     by (rule closed_Un)
   then show "closed (insert a S)"
     by simp
 qed
 
 lemma finite_imp_closed: "finite S \<Longrightarrow> closed S"
   by (induct pred: finite) simp_all
 
 end
 
 text \<open>T2 spaces are also known as Hausdorff spaces.\<close>
 
 class t2_space = topological_space +
   assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
 
 instance t2_space \<subseteq> t1_space
   by standard (fast dest: hausdorff)
 
 lemma (in t2_space) separation_t2: "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
   using hausdorff [of x y] by blast
 
 lemma (in t0_space) separation_t0: "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U))"
   using t0_space [of x y] by blast
 
 
 text \<open>A classical separation axiom for topological space, the T3 axiom -- also called regularity:
 if a point is not in a closed set, then there are open sets separating them.\<close>
 
 class t3_space = t2_space +
   assumes t3_space: "closed S \<Longrightarrow> y \<notin> S \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
 
 text \<open>A classical separation axiom for topological space, the T4 axiom -- also called normality:
 if two closed sets are disjoint, then there are open sets separating them.\<close>
 
 class t4_space = t2_space +
   assumes t4_space: "closed S \<Longrightarrow> closed T \<Longrightarrow> S \<inter> T = {} \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}"
 
 text \<open>T4 is stronger than T3, and weaker than metric.\<close>
 
 instance t4_space \<subseteq> t3_space
 proof
   fix S and y::'a assume "closed S" "y \<notin> S"
   then show "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
     using t4_space[of "{y}" S] by auto
 qed
 
 text \<open>A perfect space is a topological space with no isolated points.\<close>
 
 class perfect_space = topological_space +
   assumes not_open_singleton: "\<not> open {x}"
 
 lemma (in perfect_space) UNIV_not_singleton: "UNIV \<noteq> {x}"
   for x::'a
   by (metis (no_types) open_UNIV not_open_singleton)
 
 
 subsection \<open>Generators for toplogies\<close>
 
 inductive generate_topology :: "'a set set \<Rightarrow> 'a set \<Rightarrow> bool" for S :: "'a set set"
   where
     UNIV: "generate_topology S UNIV"
   | Int: "generate_topology S (a \<inter> b)" if "generate_topology S a" and "generate_topology S b"
   | UN: "generate_topology S (\<Union>K)" if "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k)"
   | Basis: "generate_topology S s" if "s \<in> S"
 
 hide_fact (open) UNIV Int UN Basis
 
 lemma generate_topology_Union:
   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
   using generate_topology.UN [of "K ` I"] by auto
 
 lemma topological_space_generate_topology: "class.topological_space (generate_topology S)"
   by standard (auto intro: generate_topology.intros)
 
 
 subsection \<open>Order topologies\<close>
 
 class order_topology = order + "open" +
   assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 begin
 
 subclass topological_space
   unfolding open_generated_order
   by (rule topological_space_generate_topology)
 
 lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"
   unfolding open_generated_order by (auto intro: generate_topology.Basis)
 
 lemma open_lessThan [continuous_intros, simp]: "open {..< a}"
   unfolding open_generated_order by (auto intro: generate_topology.Basis)
 
 lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}"
    unfolding greaterThanLessThan_eq by (simp add: open_Int)
 
 end
 
 class linorder_topology = linorder + order_topology
 
 lemma closed_atMost [continuous_intros, simp]: "closed {..a}"
   for a :: "'a::linorder_topology"
   by (simp add: closed_open)
 
 lemma closed_atLeast [continuous_intros, simp]: "closed {a..}"
   for a :: "'a::linorder_topology"
   by (simp add: closed_open)
 
 lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}"
   for a b :: "'a::linorder_topology"
 proof -
   have "{a .. b} = {a ..} \<inter> {.. b}"
     by auto
   then show ?thesis
     by (simp add: closed_Int)
 qed
 
 lemma (in order) less_separate:
   assumes "x < y"
   shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
 proof (cases "\<exists>z. x < z \<and> z < y")
   case True
   then obtain z where "x < z \<and> z < y" ..
   then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
     by auto
   then show ?thesis by blast
 next
   case False
   with \<open>x < y\<close> have "x \<in> {..< y}" "y \<in> {x <..}" "{x <..} \<inter> {..< y} = {}"
     by auto
   then show ?thesis by blast
 qed
 
 instance linorder_topology \<subseteq> t2_space
 proof
   fix x y :: 'a
   show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
     using less_separate [of x y] less_separate [of y x]
     by (elim neqE; metis open_lessThan open_greaterThan Int_commute)
 qed
 
 lemma (in linorder_topology) open_right:
   assumes "open S" "x \<in> S"
     and gt_ex: "x < y"
   shows "\<exists>b>x. {x ..< b} \<subseteq> S"
   using assms unfolding open_generated_order
 proof induct
   case UNIV
   then show ?case by blast
 next
   case (Int A B)
   then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B"
     by auto
   then show ?case
     by (auto intro!: exI[of _ "min a b"])
 next
   case UN
   then show ?case by blast
 next
   case Basis
   then show ?case
     by (fastforce intro: exI[of _ y] gt_ex)
 qed
 
 lemma (in linorder_topology) open_left:
   assumes "open S" "x \<in> S"
     and lt_ex: "y < x"
   shows "\<exists>b<x. {b <.. x} \<subseteq> S"
   using assms unfolding open_generated_order
 proof induction
   case UNIV
   then show ?case by blast
 next
   case (Int A B)
   then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B"
     by auto
   then show ?case
     by (auto intro!: exI[of _ "max a b"])
 next
   case UN
   then show ?case by blast
 next
   case Basis
   then show ?case
     by (fastforce intro: exI[of _ y] lt_ex)
 qed
 
 
 subsection \<open>Setup some topologies\<close>
 
 subsubsection \<open>Boolean is an order topology\<close>
 
 class discrete_topology = topological_space +
   assumes open_discrete: "\<And>A. open A"
 
 instance discrete_topology < t2_space
 proof
   fix x y :: 'a
   assume "x \<noteq> y"
   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
     by (intro exI[of _ "{_}"]) (auto intro!: open_discrete)
 qed
 
 instantiation bool :: linorder_topology
 begin
 
 definition open_bool :: "bool set \<Rightarrow> bool"
   where "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
   by standard (rule open_bool_def)
 
 end
 
 instance bool :: discrete_topology
 proof
   fix A :: "bool set"
   have *: "{False <..} = {True}" "{..< True} = {False}"
     by auto
   have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}"
     using subset_UNIV[of A] unfolding UNIV_bool * by blast
   then show "open A"
     by auto
 qed
 
 instantiation nat :: linorder_topology
 begin
 
 definition open_nat :: "nat set \<Rightarrow> bool"
   where "open_nat = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
   by standard (rule open_nat_def)
 
 end
 
 instance nat :: discrete_topology
 proof
   fix A :: "nat set"
   have "open {n}" for n :: nat
   proof (cases n)
     case 0
     moreover have "{0} = {..<1::nat}"
       by auto
     ultimately show ?thesis
        by auto
   next
     case (Suc n')
     then have "{n} = {..<Suc n} \<inter> {n' <..}"
       by auto
     with Suc show ?thesis
       by (auto intro: open_lessThan open_greaterThan)
   qed
   then have "open (\<Union>a\<in>A. {a})"
     by (intro open_UN) auto
   then show "open A"
     by simp
 qed
 
 instantiation int :: linorder_topology
 begin
 
 definition open_int :: "int set \<Rightarrow> bool"
   where "open_int = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
 
 instance
   by standard (rule open_int_def)
 
 end
 
 instance int :: discrete_topology
 proof
   fix A :: "int set"
   have "{..<i + 1} \<inter> {i-1 <..} = {i}" for i :: int
     by auto
   then have "open {i}" for i :: int
     using open_Int[OF open_lessThan[of "i + 1"] open_greaterThan[of "i - 1"]] by auto
   then have "open (\<Union>a\<in>A. {a})"
     by (intro open_UN) auto
   then show "open A"
     by simp
 qed
 
 
 subsubsection \<open>Topological filters\<close>
 
 definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
   where "nhds a = (INF S\<in>{S. open S \<and> a \<in> S}. principal S)"
 
 definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter"
     ("at (_)/ within (_)" [1000, 60] 60)
   where "at a within s = inf (nhds a) (principal (s - {a}))"
 
 abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter"  ("at")
   where "at x \<equiv> at x within (CONST UNIV)"
 
 abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter"
   where "at_right x \<equiv> at x within {x <..}"
 
 abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter"
   where "at_left x \<equiv> at x within {..< x}"
 
 lemma (in topological_space) nhds_generated_topology:
   "open = generate_topology T \<Longrightarrow> nhds x = (INF S\<in>{S\<in>T. x \<in> S}. principal S)"
   unfolding nhds_def
 proof (safe intro!: antisym INF_greatest)
   fix S
   assume "generate_topology T S" "x \<in> S"
   then show "(INF S\<in>{S \<in> T. x \<in> S}. principal S) \<le> principal S"
     by induct
       (auto intro: INF_lower order_trans simp: inf_principal[symmetric] simp del: inf_principal)
 qed (auto intro!: INF_lower intro: generate_topology.intros)
 
 lemma (in topological_space) eventually_nhds:
   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
   unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
 
 lemma eventually_eventually:
   "eventually (\<lambda>y. eventually P (nhds y)) (nhds x) = eventually P (nhds x)"
   by (auto simp: eventually_nhds)
 
 lemma (in topological_space) eventually_nhds_in_open:
   "open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
   by (subst eventually_nhds) blast
 
 lemma (in topological_space) eventually_nhds_x_imp_x: "eventually P (nhds x) \<Longrightarrow> P x"
   by (subst (asm) eventually_nhds) blast
 
 lemma (in topological_space) nhds_neq_bot [simp]: "nhds a \<noteq> bot"
   by (simp add: trivial_limit_def eventually_nhds)
 
 lemma (in t1_space) t1_space_nhds: "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
   by (drule t1_space) (auto simp: eventually_nhds)
 
 lemma (in topological_space) nhds_discrete_open: "open {x} \<Longrightarrow> nhds x = principal {x}"
   by (auto simp: nhds_def intro!: antisym INF_greatest INF_lower2[of "{x}"])
 
 lemma (in discrete_topology) nhds_discrete: "nhds x = principal {x}"
   by (simp add: nhds_discrete_open open_discrete)
 
 lemma (in discrete_topology) at_discrete: "at x within S = bot"
   unfolding at_within_def nhds_discrete by simp
 
 lemma (in discrete_topology) tendsto_discrete:
   "filterlim (f :: 'b \<Rightarrow> 'a) (nhds y) F \<longleftrightarrow> eventually (\<lambda>x. f x = y) F"
   by (auto simp: nhds_discrete filterlim_principal)
 
 lemma (in topological_space) at_within_eq:
   "at x within s = (INF S\<in>{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
   unfolding nhds_def at_within_def
   by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib)
 
 lemma (in topological_space) eventually_at_filter:
   "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
   by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute)
 
 lemma (in topological_space) at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
   unfolding at_within_def by (intro inf_mono) auto
 
 lemma (in topological_space) eventually_at_topological:
   "eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
   by (simp add: eventually_nhds eventually_at_filter)
 
 lemma (in topological_space) at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
   unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
 
 lemma (in topological_space) at_within_open_NO_MATCH:
   "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
   by (simp only: at_within_open)
 
 lemma (in topological_space) at_within_open_subset:
   "a \<in> S \<Longrightarrow> open S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> at a within T = at a"
   by (metis at_le at_within_open dual_order.antisym subset_UNIV)
 
 lemma (in topological_space) at_within_nhd:
   assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}"
   shows "at x within T = at x within U"
   unfolding filter_eq_iff eventually_at_filter
 proof (intro allI eventually_subst)
   have "eventually (\<lambda>x. x \<in> S) (nhds x)"
     using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds)
   then show "\<forall>\<^sub>F n in nhds x. (n \<noteq> x \<longrightarrow> n \<in> T \<longrightarrow> P n) = (n \<noteq> x \<longrightarrow> n \<in> U \<longrightarrow> P n)" for P
     by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast)
 qed
 
 lemma (in topological_space) at_within_empty [simp]: "at a within {} = bot"
   unfolding at_within_def by simp
 
 lemma (in topological_space) at_within_union:
   "at x within (S \<union> T) = sup (at x within S) (at x within T)"
   unfolding filter_eq_iff eventually_sup eventually_at_filter
   by (auto elim!: eventually_rev_mp)
 
 lemma (in topological_space) at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
   unfolding trivial_limit_def eventually_at_topological
   apply safe
    apply (case_tac "S = {a}")
     apply simp
    apply fast
   apply fast
   done
 
 lemma (in perfect_space) at_neq_bot [simp]: "at a \<noteq> bot"
   by (simp add: at_eq_bot_iff not_open_singleton)
 
 lemma (in order_topology) nhds_order:
   "nhds x = inf (INF a\<in>{x <..}. principal {..< a}) (INF a\<in>{..< x}. principal {a <..})"
 proof -
   have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} =
       (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
     by auto
   show ?thesis
     by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def)
 qed
 
 lemma (in topological_space) filterlim_at_within_If:
   assumes "filterlim f G (at x within (A \<inter> {x. P x}))"
     and "filterlim g G (at x within (A \<inter> {x. \<not>P x}))"
   shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x within A)"
 proof (rule filterlim_If)
   note assms(1)
   also have "at x within (A \<inter> {x. P x}) = inf (nhds x) (principal (A \<inter> Collect P - {x}))"
     by (simp add: at_within_def)
   also have "A \<inter> Collect P - {x} = (A - {x}) \<inter> Collect P"
     by blast
   also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal (Collect P))"
     by (simp add: at_within_def inf_assoc)
   finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" .
 next
   note assms(2)
   also have "at x within (A \<inter> {x. \<not> P x}) = inf (nhds x) (principal (A \<inter> {x. \<not> P x} - {x}))"
     by (simp add: at_within_def)
   also have "A \<inter> {x. \<not> P x} - {x} = (A - {x}) \<inter> {x. \<not> P x}"
     by blast
   also have "inf (nhds x) (principal \<dots>) = inf (at x within A) (principal {x. \<not> P x})"
     by (simp add: at_within_def inf_assoc)
   finally show "filterlim g G (inf (at x within A) (principal {x. \<not> P x}))" .
 qed
 
 lemma (in topological_space) filterlim_at_If:
   assumes "filterlim f G (at x within {x. P x})"
     and "filterlim g G (at x within {x. \<not>P x})"
   shows "filterlim (\<lambda>x. if P x then f x else g x) G (at x)"
   using assms by (intro filterlim_at_within_If) simp_all
 lemma (in linorder_topology) at_within_order:
   assumes "UNIV \<noteq> {x}"
   shows "at x within s =
     inf (INF a\<in>{x <..}. principal ({..< a} \<inter> s - {x}))
         (INF a\<in>{..< x}. principal ({a <..} \<inter> s - {x}))"
 proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split])
   case True_True
   have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
     by auto
   with assms True_True show ?thesis
     by auto
 qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff
       inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"])
 
 lemma (in linorder_topology) at_left_eq:
   "y < x \<Longrightarrow> at_left x = (INF a\<in>{..< x}. principal {a <..< x})"
   by (subst at_within_order)
      (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
            intro!: INF_lower2 inf_absorb2)
 
 lemma (in linorder_topology) eventually_at_left:
   "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
   unfolding at_left_eq
   by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
 
 lemma (in linorder_topology) at_right_eq:
   "x < y \<Longrightarrow> at_right x = (INF a\<in>{x <..}. principal {x <..< a})"
   by (subst at_within_order)
      (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
            intro!: INF_lower2 inf_absorb1)
 
 lemma (in linorder_topology) eventually_at_right:
   "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
   unfolding at_right_eq
   by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
 
 lemma eventually_at_right_less: "\<forall>\<^sub>F y in at_right (x::'a::{linorder_topology, no_top}). x < y"
   using gt_ex[of x] eventually_at_right[of x] by auto
 
 lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot"
   by (auto simp: filter_eq_iff eventually_at_topological)
 
 lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot"
   by (auto simp: filter_eq_iff eventually_at_topological)
 
 lemma trivial_limit_at_left_real [simp]: "\<not> trivial_limit (at_left x)"
   for x :: "'a::{no_bot,dense_order,linorder_topology}"
   using lt_ex [of x]
   by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
 
 lemma trivial_limit_at_right_real [simp]: "\<not> trivial_limit (at_right x)"
   for x :: "'a::{no_top,dense_order,linorder_topology}"
   using gt_ex[of x]
   by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
 
 lemma (in linorder_topology) at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)"
   by (auto simp: eventually_at_filter filter_eq_iff eventually_sup
       elim: eventually_elim2 eventually_mono)
 
 lemma (in linorder_topology) eventually_at_split:
   "eventually P (at x) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
   by (subst at_eq_sup_left_right) (simp add: eventually_sup)
 
 lemma (in order_topology) eventually_at_leftI:
   assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b"
   shows   "eventually P (at_left b)"
   using assms unfolding eventually_at_topological by (intro exI[of _ "{a<..}"]) auto
 
 lemma (in order_topology) eventually_at_rightI:
   assumes "\<And>x. x \<in> {a<..<b} \<Longrightarrow> P x" "a < b"
   shows   "eventually P (at_right a)"
   using assms unfolding eventually_at_topological by (intro exI[of _ "{..<b}"]) auto
 
 lemma eventually_filtercomap_nhds:
   "eventually P (filtercomap f (nhds x)) \<longleftrightarrow> (\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x. f x \<in> S \<longrightarrow> P x))"
   unfolding eventually_filtercomap eventually_nhds by auto
 
 lemma eventually_filtercomap_at_topological:
   "eventually P (filtercomap f (at A within B)) \<longleftrightarrow> 
      (\<exists>S. open S \<and> A \<in> S \<and> (\<forall>x. f x \<in> S \<inter> B - {A} \<longrightarrow> P x))" (is "?lhs = ?rhs")
   unfolding at_within_def filtercomap_inf eventually_inf_principal filtercomap_principal 
           eventually_filtercomap_nhds eventually_principal by blast
 
 lemma eventually_at_right_field:
   "eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
   for x :: "'a::{linordered_field, linorder_topology}"
   using linordered_field_no_ub[rule_format, of x]
   by (auto simp: eventually_at_right)
 
 lemma eventually_at_left_field:
   "eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
   for x :: "'a::{linordered_field, linorder_topology}"
   using linordered_field_no_lb[rule_format, of x]
   by (auto simp: eventually_at_left)
 
 
 subsubsection \<open>Tendsto\<close>
 
 abbreviation (in topological_space)
   tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool"  (infixr "\<longlongrightarrow>" 55)
   where "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
 
 definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a"
   where "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
 
 lemma (in topological_space) tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F"
   by simp
 
 named_theorems tendsto_intros "introduction rules for tendsto"
 setup \<open>
   Global_Theory.add_thms_dynamic (\<^binding>\<open>tendsto_eq_intros\<close>,
     fn context =>
       Named_Theorems.get (Context.proof_of context) \<^named_theorems>\<open>tendsto_intros\<close>
       |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
 \<close>
 
 context topological_space begin
 
 lemma tendsto_def:
    "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
    unfolding nhds_def filterlim_INF filterlim_principal by auto
 
 lemma tendsto_cong: "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F" if "eventually (\<lambda>x. f x = g x) F"
   by (rule filterlim_cong [OF refl refl that])
 
 lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F"
   unfolding tendsto_def le_filter_def by fast
 
 lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
   by (auto simp: tendsto_def eventually_at_topological)
 
 lemma tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F"
   by (simp add: tendsto_def)
 
 lemma filterlim_at:
   "(LIM x F. f x :> at b within s) \<longleftrightarrow> eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F"
   by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
 
 lemma (in -)
   assumes "filterlim f (nhds L) F"
   shows tendsto_imp_filterlim_at_right:
           "eventually (\<lambda>x. f x > L) F \<Longrightarrow> filterlim f (at_right L) F"
     and tendsto_imp_filterlim_at_left:
           "eventually (\<lambda>x. f x < L) F \<Longrightarrow> filterlim f (at_left L) F"
   using assms by (auto simp: filterlim_at elim: eventually_mono)
 
 lemma  filterlim_at_withinI:
   assumes "filterlim f (nhds c) F"
   assumes "eventually (\<lambda>x. f x \<in> A - {c}) F"
   shows   "filterlim f (at c within A) F"
   using assms by (simp add: filterlim_at)
 
 lemma filterlim_atI:
   assumes "filterlim f (nhds c) F"
   assumes "eventually (\<lambda>x. f x \<noteq> c) F"
   shows   "filterlim f (at c) F"
   using assms by (intro filterlim_at_withinI) simp_all
 
 lemma topological_tendstoI:
   "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
   by (auto simp: tendsto_def)
 
 lemma topological_tendstoD:
   "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
   by (auto simp: tendsto_def)
 
 lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot"
   by (simp add: tendsto_def)
 
 lemma tendsto_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> ((\<lambda>x. f x) \<longlongrightarrow> l) net"
   by (rule topological_tendstoI) (auto elim: eventually_mono)
 
 end
 
 lemma (in topological_space) filterlim_within_subset:
   "filterlim f l (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> filterlim f l (at x within T)"
   by (blast intro: filterlim_mono at_le)
 
 lemmas tendsto_within_subset = filterlim_within_subset
 
 lemma (in order_topology) order_tendsto_iff:
   "(f \<longlongrightarrow> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
   by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal)
 
 lemma (in order_topology) order_tendstoI:
   "(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow>
     (f \<longlongrightarrow> y) F"
   by (auto simp: order_tendsto_iff)
 
 lemma (in order_topology) order_tendstoD:
   assumes "(f \<longlongrightarrow> y) F"
   shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
     and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
   using assms by (auto simp: order_tendsto_iff)
 
 lemma (in linorder_topology) tendsto_max[tendsto_intros]:
   assumes X: "(X \<longlongrightarrow> x) net"
     and Y: "(Y \<longlongrightarrow> y) net"
   shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net"
 proof (rule order_tendstoI)
   fix a
   assume "a < max x y"
   then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
     by (auto simp: less_max_iff_disj elim: eventually_mono)
 next
   fix a
   assume "max x y < a"
   then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
     by (auto simp: eventually_conj_iff)
 qed
 
 lemma (in linorder_topology) tendsto_min[tendsto_intros]:
   assumes X: "(X \<longlongrightarrow> x) net"
     and Y: "(Y \<longlongrightarrow> y) net"
   shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
 proof (rule order_tendstoI)
   fix a
   assume "a < min x y"
   then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
     using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
     by (auto simp: eventually_conj_iff)
 next
   fix a
   assume "min x y < a"
   then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
     using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
     by (auto simp: min_less_iff_disj elim: eventually_mono)
 qed
 
 lemma (in order_topology)
   assumes "a < b"
   shows at_within_Icc_at_right: "at a within {a..b} = at_right a"
     and at_within_Icc_at_left:  "at b within {a..b} = at_left b"
   using order_tendstoD(2)[OF tendsto_ident_at assms, of "{a<..}"]
   using order_tendstoD(1)[OF tendsto_ident_at assms, of "{..<b}"]
   by (auto intro!: order_class.antisym filter_leI
       simp: eventually_at_filter less_le
       elim: eventually_elim2)
 
 lemma (in order_topology) at_within_Icc_at: "a < x \<Longrightarrow> x < b \<Longrightarrow> at x within {a..b} = at x"
   by (rule at_within_open_subset[where S="{a<..<b}"]) auto
 
 lemma (in t2_space) tendsto_unique:
   assumes "F \<noteq> bot"
     and "(f \<longlongrightarrow> a) F"
     and "(f \<longlongrightarrow> b) F"
   shows "a = b"
 proof (rule ccontr)
   assume "a \<noteq> b"
   obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
     using hausdorff [OF \<open>a \<noteq> b\<close>] by fast
   have "eventually (\<lambda>x. f x \<in> U) F"
     using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
   moreover
   have "eventually (\<lambda>x. f x \<in> V) F"
     using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD)
   ultimately
   have "eventually (\<lambda>x. False) F"
   proof eventually_elim
     case (elim x)
     then have "f x \<in> U \<inter> V" by simp
     with \<open>U \<inter> V = {}\<close> show ?case by simp
   qed
   with \<open>\<not> trivial_limit F\<close> show "False"
     by (simp add: trivial_limit_def)
 qed
 
 lemma (in t2_space) tendsto_const_iff:
   fixes a b :: 'a
   assumes "\<not> trivial_limit F"
   shows "((\<lambda>x. a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
   by (auto intro!: tendsto_unique [OF assms tendsto_const])
 
 lemma Lim_in_closed_set:
   assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) F" "F \<noteq> bot" "(f \<longlongrightarrow> l) F"
   shows "l \<in> S"
 proof (rule ccontr)
   assume "l \<notin> S"
   with \<open>closed S\<close> have "open (- S)" "l \<in> - S"
     by (simp_all add: open_Compl)
   with assms(4) have "eventually (\<lambda>x. f x \<in> - S) F"
     by (rule topological_tendstoD)
   with assms(2) have "eventually (\<lambda>x. False) F"
     by (rule eventually_elim2) simp
   with assms(3) show "False"
     by (simp add: eventually_False)
 qed
 
 lemma (in t3_space) nhds_closed:
   assumes "x \<in> A" and "open A"
   shows   "\<exists>A'. x \<in> A' \<and> closed A' \<and> A' \<subseteq> A \<and> eventually (\<lambda>y. y \<in> A') (nhds x)"
 proof -
   from assms have "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> - A \<subseteq> V \<and> U \<inter> V = {}"
     by (intro t3_space) auto
   then obtain U V where UV: "open U" "open V" "x \<in> U" "-A \<subseteq> V" "U \<inter> V = {}"
     by auto
   have "eventually (\<lambda>y. y \<in> U) (nhds x)"
     using \<open>open U\<close> and \<open>x \<in> U\<close> by (intro eventually_nhds_in_open)
   hence "eventually (\<lambda>y. y \<in> -V) (nhds x)"
     by eventually_elim (use UV in auto)
   with UV show ?thesis by (intro exI[of _ "-V"]) auto
 qed
 
 lemma (in order_topology) increasing_tendsto:
   assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
     and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
   shows "(f \<longlongrightarrow> l) F"
   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
 
 lemma (in order_topology) decreasing_tendsto:
   assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
     and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
   shows "(f \<longlongrightarrow> l) F"
   using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
 
 lemma (in order_topology) tendsto_sandwich:
   assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
   assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net"
   shows "(g \<longlongrightarrow> c) net"
 proof (rule order_tendstoI)
   fix a
   show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
     using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
 next
   fix a
   show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
     using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
 qed
 
 lemma (in t1_space) limit_frequently_eq:
   assumes "F \<noteq> bot"
     and "frequently (\<lambda>x. f x = c) F"
     and "(f \<longlongrightarrow> d) F"
   shows "d = c"
 proof (rule ccontr)
   assume "d \<noteq> c"
   from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U"
     by blast
   with assms have "eventually (\<lambda>x. f x \<in> U) F"
     unfolding tendsto_def by blast
   then have "eventually (\<lambda>x. f x \<noteq> c) F"
     by eventually_elim (insert \<open>c \<notin> U\<close>, blast)
   with assms(2) show False
     unfolding frequently_def by contradiction
 qed
 
 lemma (in t1_space) tendsto_imp_eventually_ne:
   assumes  "(f \<longlongrightarrow> c) F" "c \<noteq> c'"
   shows "eventually (\<lambda>z. f z \<noteq> c') F"
 proof (cases "F=bot")
   case True
   thus ?thesis by auto
 next
   case False
   show ?thesis
   proof (rule ccontr)
     assume "\<not> eventually (\<lambda>z. f z \<noteq> c') F"
     then have "frequently (\<lambda>z. f z = c') F"
       by (simp add: frequently_def)
     from limit_frequently_eq[OF False this \<open>(f \<longlongrightarrow> c) F\<close>] and \<open>c \<noteq> c'\<close> show False
       by contradiction
   qed
 qed
 
 lemma (in linorder_topology) tendsto_le:
   assumes F: "\<not> trivial_limit F"
     and x: "(f \<longlongrightarrow> x) F"
     and y: "(g \<longlongrightarrow> y) F"
     and ev: "eventually (\<lambda>x. g x \<le> f x) F"
   shows "y \<le> x"
 proof (rule ccontr)
   assume "\<not> y \<le> x"
   with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
     by (auto simp: not_le)
   then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
     using x y by (auto intro: order_tendstoD)
   with ev have "eventually (\<lambda>x. False) F"
     by eventually_elim (insert xy, fastforce)
   with F show False
     by (simp add: eventually_False)
 qed
 
 lemma (in linorder_topology) tendsto_lowerbound:
   assumes x: "(f \<longlongrightarrow> x) F"
       and ev: "eventually (\<lambda>i. a \<le> f i) F"
       and F: "\<not> trivial_limit F"
   shows "a \<le> x"
   using F x tendsto_const ev by (rule tendsto_le)
 
 lemma (in linorder_topology) tendsto_upperbound:
   assumes x: "(f \<longlongrightarrow> x) F"
       and ev: "eventually (\<lambda>i. a \<ge> f i) F"
       and F: "\<not> trivial_limit F"
   shows "a \<ge> x"
   by (rule tendsto_le [OF F tendsto_const x ev])
 
 lemma filterlim_at_within_not_equal:
   fixes f::"'a \<Rightarrow> 'b::t2_space"
   assumes "filterlim f (at a within s) F"
   shows "eventually (\<lambda>w. f w\<in>s \<and> f w \<noteq>b) F"
 proof (cases "a=b")
   case True
   then show ?thesis using assms by (simp add: filterlim_at)
 next
   case False
   from hausdorff[OF this] obtain U V where UV:"open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
     by auto  
   have "(f \<longlongrightarrow> a) F" using assms filterlim_at by auto
   then have "\<forall>\<^sub>F x in F. f x \<in> U" using UV unfolding tendsto_def by auto
   moreover have  "\<forall>\<^sub>F x in F. f x \<in> s \<and> f x\<noteq>a" using assms filterlim_at by auto
   ultimately show ?thesis 
     apply eventually_elim
     using UV by auto
 qed
 
 subsubsection \<open>Rules about \<^const>\<open>Lim\<close>\<close>
 
 lemma tendsto_Lim: "\<not> trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
   unfolding Lim_def using tendsto_unique [of net f] by auto
 
 lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
   by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
 
 lemma eventually_Lim_ident_at:
   "(\<forall>\<^sub>F y in at x within X. P (Lim (at x within X) (\<lambda>x. x)) y) \<longleftrightarrow>
     (\<forall>\<^sub>F y in at x within X. P x y)" for x::"'a::t2_space"
   by (cases "at x within X = bot") (auto simp: Lim_ident_at)
 
 lemma filterlim_at_bot_at_right:
   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
     and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
     and Q: "eventually Q (at_right a)"
     and bound: "\<And>b. Q b \<Longrightarrow> a < b"
     and P: "eventually P at_bot"
   shows "filterlim f at_bot (at_right a)"
 proof -
   from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
     unfolding eventually_at_bot_linorder by auto
   show ?thesis
   proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
     fix z
     assume "z \<le> x"
     with x have "P z" by auto
     have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
       using bound[OF bij(2)[OF \<open>P z\<close>]]
       unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
       by (auto intro!: exI[of _ "g z"])
     with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
       by eventually_elim (metis bij \<open>P z\<close> mono)
   qed
 qed
 
 lemma filterlim_at_top_at_left:
   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
   assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
     and bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
     and Q: "eventually Q (at_left a)"
     and bound: "\<And>b. Q b \<Longrightarrow> b < a"
     and P: "eventually P at_top"
   shows "filterlim f at_top (at_left a)"
 proof -
   from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
     unfolding eventually_at_top_linorder by auto
   show ?thesis
   proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
     fix z
     assume "x \<le> z"
     with x have "P z" by auto
     have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
       using bound[OF bij(2)[OF \<open>P z\<close>]]
       unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]]
       by (auto intro!: exI[of _ "g z"])
     with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
       by eventually_elim (metis bij \<open>P z\<close> mono)
   qed
 qed
 
 lemma filterlim_split_at:
   "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow>
     filterlim f F (at x)"
   for x :: "'a::linorder_topology"
   by (subst at_eq_sup_left_right) (rule filterlim_sup)
 
 lemma filterlim_at_split:
   "filterlim f F (at x) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
   for x :: "'a::linorder_topology"
   by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
 
 lemma eventually_nhds_top:
   fixes P :: "'a :: {order_top,linorder_topology} \<Rightarrow> bool"
     and b :: 'a
   assumes "b < top"
   shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
   unfolding eventually_nhds
 proof safe
   fix S :: "'a set"
   assume "open S" "top \<in> S"
   note open_left[OF this \<open>b < top\<close>]
   moreover assume "\<forall>s\<in>S. P s"
   ultimately show "\<exists>b<top. \<forall>z>b. P z"
     by (auto simp: subset_eq Ball_def)
 next
   fix b
   assume "b < top" "\<forall>z>b. P z"
   then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
     by (intro exI[of _ "{b <..}"]) auto
 qed
 
 lemma tendsto_at_within_iff_tendsto_nhds:
   "(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
   unfolding tendsto_def eventually_at_filter eventually_inf_principal
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
 
 subsection \<open>Limits on sequences\<close>
 
 abbreviation (in topological_space)
   LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"  ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60)
   where "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
 
 abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"
   where "lim X \<equiv> Lim sequentially X"
 
 definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
   where "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
 
 lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)"
   unfolding Lim_def ..
 
 lemma lim_explicit:
   "f \<longlonglongrightarrow> f0 \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> f0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> S))"
   unfolding tendsto_def eventually_sequentially by auto
 
 
 subsection \<open>Monotone sequences and subsequences\<close>
 
 text \<open>
   Definition of monotonicity.
   The use of disjunction here complicates proofs considerably.
   One alternative is to add a Boolean argument to indicate the direction.
   Another is to develop the notions of increasing and decreasing first.
 \<close>
 definition monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
   where "monoseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
 
 abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
   where "incseq X \<equiv> mono X"
 
 lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
   unfolding mono_def ..
 
 abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool"
   where "decseq X \<equiv> antimono X"
 
 lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
   unfolding antimono_def ..
 
 subsubsection \<open>Definition of subsequence.\<close>
 
 (* For compatibility with the old "subseq" *)
 lemma strict_mono_leD: "strict_mono r \<Longrightarrow> m \<le> n \<Longrightarrow> r m \<le> r n"
   by (erule (1) monoD [OF strict_mono_mono])
 
 lemma strict_mono_id: "strict_mono id"
   by (simp add: strict_mono_def)
 
 lemma incseq_SucI: "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
   using lift_Suc_mono_le[of X] by (auto simp: incseq_def)
 
 lemma incseqD: "incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
   by (auto simp: incseq_def)
 
 lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
   using incseqD[of A i "Suc i"] by auto
 
 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
   by (auto intro: incseq_SucI dest: incseq_SucD)
 
 lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
   unfolding incseq_def by auto
 
 lemma decseq_SucI: "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
   using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def)
 
 lemma decseqD: "decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
   by (auto simp: decseq_def)
 
 lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
   using decseqD[of A i "Suc i"] by auto
 
 lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
   by (auto intro: decseq_SucI dest: decseq_SucD)
 
 lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
   unfolding decseq_def by auto
 
 lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
   unfolding monoseq_def incseq_def decseq_def ..
 
 lemma monoseq_Suc: "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
   unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
 
 lemma monoI1: "\<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> monoseq X"
   by (simp add: monoseq_def)
 
 lemma monoI2: "\<forall>m. \<forall>n \<ge> m. X n \<le> X m \<Longrightarrow> monoseq X"
   by (simp add: monoseq_def)
 
 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) \<Longrightarrow> monoseq X"
   by (simp add: monoseq_Suc)
 
 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n \<Longrightarrow> monoseq X"
   by (simp add: monoseq_Suc)
 
 lemma monoseq_minus:
   fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
   assumes "monoseq a"
   shows "monoseq (\<lambda> n. - a n)"
 proof (cases "\<forall>m. \<forall>n \<ge> m. a m \<le> a n")
   case True
   then have "\<forall>m. \<forall>n \<ge> m. - a n \<le> - a m" by auto
   then show ?thesis by (rule monoI2)
 next
   case False
   then have "\<forall>m. \<forall>n \<ge> m. - a m \<le> - a n"
     using \<open>monoseq a\<close>[unfolded monoseq_def] by auto
   then show ?thesis by (rule monoI1)
 qed
 
 
 subsubsection \<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>
 
 lemma strict_mono_Suc_iff: "strict_mono f \<longleftrightarrow> (\<forall>n. f n < f (Suc n))"
 proof (intro iffI strict_monoI)
   assume *: "\<forall>n. f n < f (Suc n)"
   fix m n :: nat assume "m < n"
   thus "f m < f n"
     by (induction rule: less_Suc_induct) (use * in auto)
 qed (auto simp: strict_mono_def)
 
 lemma strict_mono_add: "strict_mono (\<lambda>n::'a::linordered_semidom. n + k)"
   by (auto simp: strict_mono_def)
 
 text \<open>For any sequence, there is a monotonic subsequence.\<close>
 lemma seq_monosub:
   fixes s :: "nat \<Rightarrow> 'a::linorder"
   shows "\<exists>f. strict_mono f \<and> monoseq (\<lambda>n. (s (f n)))"
 proof (cases "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p")
   case True
   then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
     by (intro dependent_nat_choice) (auto simp: conj_commute)
   then obtain f :: "nat \<Rightarrow> nat" 
     where f: "strict_mono f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
     by (auto simp: strict_mono_Suc_iff)
   then have "incseq f"
     unfolding strict_mono_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
   then have "monoseq (\<lambda>n. s (f n))"
     by (auto simp add: incseq_def intro!: mono monoI2)
   with f show ?thesis
     by auto
 next
   case False
   then obtain N where N: "p > N \<Longrightarrow> \<exists>m>p. s p < s m" for p
     by (force simp: not_le le_less)
   have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
   proof (intro dependent_nat_choice)
     fix x
     assume "N < x" with N[of x]
     show "\<exists>y>N. x < y \<and> s x \<le> s y"
       by (auto intro: less_trans)
   qed auto
   then show ?thesis
     by (auto simp: monoseq_iff incseq_Suc_iff strict_mono_Suc_iff)
 qed
 
 lemma seq_suble:
   assumes sf: "strict_mono (f :: nat \<Rightarrow> nat)"
   shows "n \<le> f n"
 proof (induct n)
   case 0
   show ?case by simp
 next
   case (Suc n)
   with sf [unfolded strict_mono_Suc_iff, rule_format, of n] have "n < f (Suc n)"
      by arith
   then show ?case by arith
 qed
 
 lemma eventually_subseq:
   "strict_mono r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
   unfolding eventually_sequentially by (metis seq_suble le_trans)
 
 lemma not_eventually_sequentiallyD:
   assumes "\<not> eventually P sequentially"
   shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. \<not> P (r n))"
 proof -
   from assms have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
     unfolding eventually_sequentially by (simp add: not_less)
   then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
     by (auto simp: choice_iff)
   then show ?thesis
     by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
              simp: less_eq_Suc_le strict_mono_Suc_iff)
 qed
 
 lemma sequentially_offset: 
   assumes "eventually (\<lambda>i. P i) sequentially"
   shows "eventually (\<lambda>i. P (i + k)) sequentially"
   using assms by (rule eventually_sequentially_seg [THEN iffD2])
 
 lemma seq_offset_neg: 
   "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) \<longlongrightarrow> l) sequentially"
   apply (erule filterlim_compose)
   apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith)
   done
 
 lemma filterlim_subseq: "strict_mono f \<Longrightarrow> filterlim f sequentially sequentially"
   unfolding filterlim_iff by (metis eventually_subseq)
 
 lemma strict_mono_o: "strict_mono r \<Longrightarrow> strict_mono s \<Longrightarrow> strict_mono (r \<circ> s)"
   unfolding strict_mono_def by simp
 
 lemma strict_mono_compose: "strict_mono r \<Longrightarrow> strict_mono s \<Longrightarrow> strict_mono (\<lambda>x. r (s x))"
   using strict_mono_o[of r s] by (simp add: o_def)
 
 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
   by (simp add: incseq_def monoseq_def)
 
 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
   by (simp add: decseq_def monoseq_def)
 
 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)"
   for X :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
   by (simp add: decseq_def incseq_def)
 
 lemma INT_decseq_offset:
   assumes "decseq F"
   shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
 proof safe
   fix x i
   assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
   show "x \<in> F i"
   proof cases
     from x have "x \<in> F n" by auto
     also assume "i \<le> n" with \<open>decseq F\<close> have "F n \<subseteq> F i"
       unfolding decseq_def by simp
     finally show ?thesis .
   qed (insert x, simp)
 qed auto
 
 lemma LIMSEQ_const_iff: "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
   for k l :: "'a::t2_space"
   using trivial_limit_sequentially by (rule tendsto_const_iff)
 
 lemma LIMSEQ_SUP: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a::{complete_linorder,linorder_topology})"
   by (intro increasing_tendsto)
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
 
 lemma LIMSEQ_INF: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a::{complete_linorder,linorder_topology})"
   by (intro decreasing_tendsto)
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
 
 lemma LIMSEQ_ignore_initial_segment: "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
   unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k])
 
 lemma LIMSEQ_offset: "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
   unfolding tendsto_def
   by (subst (asm) eventually_sequentially_seg[where k=k])
 
 lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
   by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp
 
 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l"
   by (rule LIMSEQ_offset [where k="Suc 0"]) simp
 
 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l"
   by (rule filterlim_sequentially_Suc)
 
 lemma LIMSEQ_lessThan_iff_atMost:
   shows "(\<lambda>n. f {..<n}) \<longlonglongrightarrow> x \<longleftrightarrow> (\<lambda>n. f {..n}) \<longlonglongrightarrow> x"
   apply (subst LIMSEQ_Suc_iff [symmetric])
   apply (simp only: lessThan_Suc_atMost)
   done
 
 lemma LIMSEQ_unique: "X \<longlonglongrightarrow> a \<Longrightarrow> X \<longlonglongrightarrow> b \<Longrightarrow> a = b"
   for a b :: "'a::t2_space"
   using trivial_limit_sequentially by (rule tendsto_unique)
 
 lemma LIMSEQ_le_const: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. a \<le> X n \<Longrightarrow> a \<le> x"
   for a x :: "'a::linorder_topology"
   by (simp add: eventually_at_top_linorder tendsto_lowerbound)
 
 lemma LIMSEQ_le: "X \<longlonglongrightarrow> x \<Longrightarrow> Y \<longlonglongrightarrow> y \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> Y n \<Longrightarrow> x \<le> y"
   for x y :: "'a::linorder_topology"
   using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
 
 lemma LIMSEQ_le_const2: "X \<longlonglongrightarrow> x \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X n \<le> a \<Longrightarrow> x \<le> a"
   for a x :: "'a::linorder_topology"
   by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
 
 lemma Lim_bounded: "f \<longlonglongrightarrow> l \<Longrightarrow> \<forall>n\<ge>M. f n \<le> C \<Longrightarrow> l \<le> C"
   for l :: "'a::linorder_topology"
   by (intro LIMSEQ_le_const2) auto
 
 lemma Lim_bounded2:
   fixes f :: "nat \<Rightarrow> 'a::linorder_topology"
   assumes lim:"f \<longlonglongrightarrow> l" and ge: "\<forall>n\<ge>N. f n \<ge> C"
   shows "l \<ge> C"
   using ge
   by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const])
      (auto simp: eventually_sequentially)
 
 lemma lim_mono:
   fixes X Y :: "nat \<Rightarrow> 'a::linorder_topology"
   assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n"
     and "X \<longlonglongrightarrow> x"
     and "Y \<longlonglongrightarrow> y"
   shows "x \<le> y"
   using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto
 
 lemma Sup_lim:
   fixes a :: "'a::{complete_linorder,linorder_topology}"
   assumes "\<And>n. b n \<in> s"
     and "b \<longlonglongrightarrow> a"
   shows "a \<le> Sup s"
   by (metis Lim_bounded assms complete_lattice_class.Sup_upper)
 
 lemma Inf_lim:
   fixes a :: "'a::{complete_linorder,linorder_topology}"
   assumes "\<And>n. b n \<in> s"
     and "b \<longlonglongrightarrow> a"
   shows "Inf s \<le> a"
   by (metis Lim_bounded2 assms complete_lattice_class.Inf_lower)
 
 lemma SUP_Lim:
   fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
   assumes inc: "incseq X"
     and l: "X \<longlonglongrightarrow> l"
   shows "(SUP n. X n) = l"
   using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l]
   by simp
 
 lemma INF_Lim:
   fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
   assumes dec: "decseq X"
     and l: "X \<longlonglongrightarrow> l"
   shows "(INF n. X n) = l"
   using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l]
   by simp
 
 lemma convergentD: "convergent X \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L"
   by (simp add: convergent_def)
 
 lemma convergentI: "X \<longlonglongrightarrow> L \<Longrightarrow> convergent X"
   by (auto simp add: convergent_def)
 
 lemma convergent_LIMSEQ_iff: "convergent X \<longleftrightarrow> X \<longlonglongrightarrow> lim X"
   by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
 
 lemma convergent_const: "convergent (\<lambda>n. c)"
   by (rule convergentI) (rule tendsto_const)
 
 lemma monoseq_le:
   "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> x \<Longrightarrow>
     (\<forall>n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n) \<or>
     (\<forall>n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)"
   for x :: "'a::linorder_topology"
   by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
 
 lemma LIMSEQ_subseq_LIMSEQ: "X \<longlonglongrightarrow> L \<Longrightarrow> strict_mono f \<Longrightarrow> (X \<circ> f) \<longlonglongrightarrow> L"
   unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq])
 
 lemma convergent_subseq_convergent: "convergent X \<Longrightarrow> strict_mono f \<Longrightarrow> convergent (X \<circ> f)"
   by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ)
 
 lemma limI: "X \<longlonglongrightarrow> L \<Longrightarrow> lim X = L"
   by (rule tendsto_Lim) (rule trivial_limit_sequentially)
 
 lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> x) \<Longrightarrow> lim f \<le> x"
   for x :: "'a::linorder_topology"
   using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
 
 lemma lim_const [simp]: "lim (\<lambda>m. a) = a"
   by (simp add: limI)
 
 
 subsubsection \<open>Increasing and Decreasing Series\<close>
 
 lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> L"
   for L :: "'a::linorder_topology"
   by (metis incseq_def LIMSEQ_le_const)
 
 lemma decseq_ge: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> L \<le> X n"
   for L :: "'a::linorder_topology"
   by (metis decseq_def LIMSEQ_le_const2)
 
 
 subsection \<open>First countable topologies\<close>
 
 class first_countable_topology = topological_space +
   assumes first_countable_basis:
     "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
 
 lemma (in first_countable_topology) countable_basis_at_decseq:
   obtains A :: "nat \<Rightarrow> 'a set" where
     "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
 proof atomize_elim
   from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set"
     where nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
       and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
     by auto
   define F where "F n = (\<Inter>i\<le>n. A i)" for n
   show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
     (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
   proof (safe intro!: exI[of _ F])
     fix i
     show "open (F i)"
       using nhds(1) by (auto simp: F_def)
     show "x \<in> F i"
       using nhds(2) by (auto simp: F_def)
   next
     fix S
     assume "open S" "x \<in> S"
     from incl[OF this] obtain i where "F i \<subseteq> S"
       unfolding F_def by auto
     moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
       by (simp add: Inf_superset_mono F_def image_mono)
     ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
       by (auto simp: eventually_sequentially)
   qed
 qed
 
 lemma (in first_countable_topology) nhds_countable:
   obtains X :: "nat \<Rightarrow> 'a set"
   where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
 proof -
   from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
     where *: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
     by metis
   show thesis
   proof
     show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
       by (simp add: antimono_iff_le_Suc atMost_Suc)
     show "x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)" for n
       using * by auto
     show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))"
       using *
       unfolding nhds_def
       apply -
       apply (rule INF_eq)
        apply simp_all
        apply fastforce
       apply (intro exI [of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT)
          apply auto
       done
   qed
 qed
 
 lemma (in first_countable_topology) countable_basis:
   obtains A :: "nat \<Rightarrow> 'a set" where
     "\<And>i. open (A i)" "\<And>i. x \<in> A i"
     "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
 proof atomize_elim
   obtain A :: "nat \<Rightarrow> 'a set" where *:
     "\<And>i. open (A i)"
     "\<And>i. x \<in> A i"
     "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
     by (rule countable_basis_at_decseq) blast
   have "eventually (\<lambda>n. F n \<in> S) sequentially"
     if "\<forall>n. F n \<in> A n" "open S" "x \<in> S" for F S
     using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq)
   with * show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)"
     by (intro exI[of _ A]) (auto simp: tendsto_def)
 qed
 
 lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
   assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (inf (nhds a) (principal s))"
 proof (rule ccontr)
   obtain A :: "nat \<Rightarrow> 'a set" where *:
     "\<And>i. open (A i)"
     "\<And>i. a \<in> A i"
     "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a"
     by (rule countable_basis) blast
   assume "\<not> ?thesis"
   with * have "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
     unfolding eventually_inf_principal eventually_nhds
     by (intro choice) fastforce
   then obtain F where F: "\<forall>n. F n \<in> s" and "\<forall>n. F n \<in> A n" and F': "\<forall>n. \<not> P (F n)"
     by blast
   with * have "F \<longlonglongrightarrow> a"
     by auto
   then have "eventually (\<lambda>n. P (F n)) sequentially"
     using assms F by simp
   then show False
     by (simp add: F')
 qed
 
 lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
   "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow>
     (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
 proof (safe intro!: sequentially_imp_eventually_nhds_within)
   assume "eventually P (inf (nhds a) (principal s))"
   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
     by (auto simp: eventually_inf_principal eventually_nhds)
   moreover
   fix f
   assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
   ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
     by (auto dest!: topological_tendstoD elim: eventually_mono)
 qed
 
 lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
   "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
   using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
 
 (*Thanks to Sébastien Gouëzel*)
 lemma Inf_as_limit:
   fixes A::"'a::{linorder_topology, first_countable_topology, complete_linorder} set"
   assumes "A \<noteq> {}"
   shows "\<exists>u. (\<forall>n. u n \<in> A) \<and> u \<longlonglongrightarrow> Inf A"
 proof (cases "Inf A \<in> A")
   case True
   show ?thesis
     by (rule exI[of _ "\<lambda>n. Inf A"], auto simp add: True)
 next
   case False
   obtain y where "y \<in> A" using assms by auto
   then have "Inf A < y" using False Inf_lower less_le by auto
   obtain F :: "nat \<Rightarrow> 'a set" where F: "\<And>i. open (F i)" "\<And>i. Inf A \<in> F i"
                                        "\<And>u. (\<forall>n. u n \<in> F n) \<Longrightarrow> u \<longlonglongrightarrow> Inf A"
     by (metis first_countable_topology_class.countable_basis)
   define u where "u = (\<lambda>n. SOME z. z \<in> F n \<and> z \<in> A)"
   have "\<exists>z. z \<in> U \<and> z \<in> A" if "Inf A \<in> U" "open U" for U
   proof -
     obtain b where "b > Inf A" "{Inf A ..<b} \<subseteq> U"
       using open_right[OF \<open>open U\<close> \<open>Inf A \<in> U\<close> \<open>Inf A < y\<close>] by auto
     obtain z where "z < b" "z \<in> A"
       using \<open>Inf A < b\<close> Inf_less_iff by auto
     then have "z \<in> {Inf A ..<b}"
       by (simp add: Inf_lower)
     then show ?thesis using \<open>z \<in> A\<close> \<open>{Inf A ..<b} \<subseteq> U\<close> by auto
   qed
   then have *: "u n \<in> F n \<and> u n \<in> A" for n
     using \<open>Inf A \<in> F n\<close> \<open>open (F n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex)
   then have "u \<longlonglongrightarrow> Inf A" using F(3) by simp
   then show ?thesis using * by auto
 qed
 
 lemma tendsto_at_iff_sequentially:
   "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
   for f :: "'a::first_countable_topology \<Rightarrow> _"
   unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap
     at_within_def eventually_nhds_within_iff_sequentially comp_def
   by metis
 
 lemma approx_from_above_dense_linorder:
   fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}"
   assumes "x < y"
   shows "\<exists>u. (\<forall>n. u n > x) \<and> (u \<longlonglongrightarrow> x)"
 proof -
   obtain A :: "nat \<Rightarrow> 'a set" where A: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
                                       "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
     by (metis first_countable_topology_class.countable_basis)
   define u where "u = (\<lambda>n. SOME z. z \<in> A n \<and> z > x)"
   have "\<exists>z. z \<in> U \<and> x < z" if "x \<in> U" "open U" for U
     using open_right[OF \<open>open U\<close> \<open>x \<in> U\<close> \<open>x < y\<close>]
     by (meson atLeastLessThan_iff dense less_imp_le subset_eq)
   then have *: "u n \<in> A n \<and> x < u n" for n
     using \<open>x \<in> A n\<close> \<open>open (A n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex)
   then have "u \<longlonglongrightarrow> x" using A(3) by simp
   then show ?thesis using * by auto
 qed
 
 lemma approx_from_below_dense_linorder:
   fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}"
   assumes "x > y"
   shows "\<exists>u. (\<forall>n. u n < x) \<and> (u \<longlonglongrightarrow> x)"
 proof -
   obtain A :: "nat \<Rightarrow> 'a set" where A: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
                                       "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
     by (metis first_countable_topology_class.countable_basis)
   define u where "u = (\<lambda>n. SOME z. z \<in> A n \<and> z < x)"
   have "\<exists>z. z \<in> U \<and> z < x" if "x \<in> U" "open U" for U
     using open_left[OF \<open>open U\<close> \<open>x \<in> U\<close> \<open>x > y\<close>]
     by (meson dense greaterThanAtMost_iff less_imp_le subset_eq)
   then have *: "u n \<in> A n \<and> u n < x" for n
     using \<open>x \<in> A n\<close> \<open>open (A n)\<close> unfolding u_def by (metis (no_types, lifting) someI_ex)
   then have "u \<longlonglongrightarrow> x" using A(3) by simp
   then show ?thesis using * by auto
 qed
 
 
 subsection \<open>Function limit at a point\<close>
 
 abbreviation LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
     ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60)
   where "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)"
 
 lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)"
   by (simp add: tendsto_def at_within_open[where S = S])
 
 lemma tendsto_within_open_NO_MATCH:
   "a \<in> S \<Longrightarrow> NO_MATCH UNIV S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l)(at a)"
   for f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   using tendsto_within_open by blast
 
 lemma LIM_const_not_eq[tendsto_intros]: "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
   for a :: "'a::perfect_space" and k L :: "'b::t2_space"
   by (simp add: tendsto_const_iff)
 
 lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
 
 lemma LIM_const_eq: "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L"
   for a :: "'a::perfect_space" and k L :: "'b::t2_space"
   by (simp add: tendsto_const_iff)
 
 lemma LIM_unique: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M"
   for a :: "'a::perfect_space" and L M :: "'b::t2_space"
   using at_neq_bot by (rule tendsto_unique)
 
 
 text \<open>Limits are equal for functions equal except at limit point.\<close>
 lemma LIM_equal: "\<forall>x. x \<noteq> a \<longrightarrow> f x = g x \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)"
   by (simp add: tendsto_def eventually_at_topological)
 
 lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>b\<rightarrow> m)"
   by (simp add: LIM_equal)
 
 lemma tendsto_cong_limit: "(f \<longlongrightarrow> l) F \<Longrightarrow> k = l \<Longrightarrow> (f \<longlongrightarrow> k) F"
   by simp
 
 lemma tendsto_at_iff_tendsto_nhds: "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
   unfolding tendsto_def eventually_at_filter
   by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
 
 lemma tendsto_compose: "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
   unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
 
 lemma tendsto_compose_eventually:
   "g \<midarrow>l\<rightarrow> m \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> m) F"
   by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
 
 lemma LIM_compose_eventually:
   assumes "f \<midarrow>a\<rightarrow> b"
     and "g \<midarrow>b\<rightarrow> c"
     and "eventually (\<lambda>x. f x \<noteq> b) (at a)"
   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
   using assms(2,1,3) by (rule tendsto_compose_eventually)
 
 lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
   by (simp add: filterlim_def filtermap_filtermap comp_def)
 
 lemma tendsto_compose_at:
   assumes f: "(f \<longlongrightarrow> y) F" and g: "(g \<longlongrightarrow> z) (at y)" and fg: "eventually (\<lambda>w. f w = y \<longrightarrow> g y = z) F"
   shows "((g \<circ> f) \<longlongrightarrow> z) F"
 proof -
   have "(\<forall>\<^sub>F a in F. f a \<noteq> y) \<or> g y = z"
     using fg by force
   moreover have "(g \<longlongrightarrow> z) (filtermap f F) \<or> \<not> (\<forall>\<^sub>F a in F. f a \<noteq> y)"
     by (metis (no_types) filterlim_atI filterlim_def tendsto_mono f g)
   ultimately show ?thesis
     by (metis (no_types) f filterlim_compose filterlim_filtermap g tendsto_at_iff_tendsto_nhds tendsto_compose_filtermap)
 qed
 
 
 subsubsection \<open>Relation of \<open>LIM\<close> and \<open>LIMSEQ\<close>\<close>
 
 lemma (in first_countable_topology) sequentially_imp_eventually_within:
   "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
     eventually P (at a within s)"
   unfolding at_within_def
   by (intro sequentially_imp_eventually_nhds_within) auto
 
 lemma (in first_countable_topology) sequentially_imp_eventually_at:
   "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
   using sequentially_imp_eventually_within [where s=UNIV] by simp
 
 lemma LIMSEQ_SEQ_conv1:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   assumes f: "f \<midarrow>a\<rightarrow> l"
   shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
   using tendsto_compose_eventually [OF f, where F=sequentially] by simp
 
 lemma LIMSEQ_SEQ_conv2:
   fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
   assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
   shows "f \<midarrow>a\<rightarrow> l"
   using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
 
 lemma LIMSEQ_SEQ_conv: "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) \<longleftrightarrow> X \<midarrow>a\<rightarrow> L"
   for a :: "'a::first_countable_topology" and L :: "'b::topological_space"
   using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
 
 lemma sequentially_imp_eventually_at_left:
   fixes a :: "'a::{linorder_topology,first_countable_topology}"
   assumes b[simp]: "b < a"
     and *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
       eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (at_left a)"
 proof (safe intro!: sequentially_imp_eventually_within)
   fix X
   assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
   show "eventually (\<lambda>n. P (X n)) sequentially"
   proof (rule ccontr)
     assume neg: "\<not> ?thesis"
     have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))"
       (is "\<exists>s. ?P s")
     proof (rule dependent_nat_choice)
       have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
         by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
       then show "\<exists>x. \<not> P (X x) \<and> b < X x"
         by (auto dest!: not_eventuallyD)
     next
       fix x n
       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
         using X
         by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
       then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
         by (auto dest!: not_eventuallyD)
     qed
     then obtain s where "?P s" ..
     with X have "b < X (s n)"
       and "X (s n) < a"
       and "incseq (\<lambda>n. X (s n))"
       and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
       and "\<not> P (X (s n))"
       for n
       by (auto simp: strict_mono_Suc_iff Suc_le_eq incseq_Suc_iff
           intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
     from *[OF this(1,2,3,4)] this(5) show False
       by auto
   qed
 qed
 
 lemma tendsto_at_left_sequentially:
   fixes a b :: "'b::{linorder_topology,first_countable_topology}"
   assumes "b < a"
   assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
     (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at_left a)"
   using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left)
 
 lemma sequentially_imp_eventually_at_right:
   fixes a b :: "'a::{linorder_topology,first_countable_topology}"
   assumes b[simp]: "a < b"
   assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow>
     eventually (\<lambda>n. P (f n)) sequentially"
   shows "eventually P (at_right a)"
 proof (safe intro!: sequentially_imp_eventually_within)
   fix X
   assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
   show "eventually (\<lambda>n. P (X n)) sequentially"
   proof (rule ccontr)
     assume neg: "\<not> ?thesis"
     have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))"
       (is "\<exists>s. ?P s")
     proof (rule dependent_nat_choice)
       have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
         by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
       then show "\<exists>x. \<not> P (X x) \<and> X x < b"
         by (auto dest!: not_eventuallyD)
     next
       fix x n
       have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
         using X
         by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
       then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
         by (auto dest!: not_eventuallyD)
     qed
     then obtain s where "?P s" ..
     with X have "a < X (s n)"
       and "X (s n) < b"
       and "decseq (\<lambda>n. X (s n))"
       and "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a"
       and "\<not> P (X (s n))"
       for n
       by (auto simp: strict_mono_Suc_iff Suc_le_eq decseq_Suc_iff
           intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
     from *[OF this(1,2,3,4)] this(5) show False
       by auto
   qed
 qed
 
 lemma tendsto_at_right_sequentially:
   fixes a :: "_ :: {linorder_topology, first_countable_topology}"
   assumes "a < b"
     and *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow>
       (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
   shows "(X \<longlongrightarrow> L) (at_right a)"
   using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right)
 
 
 subsection \<open>Continuity\<close>
 
 subsubsection \<open>Continuity on a set\<close>
 
 definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
   where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
 
 lemma continuous_on_cong [cong]:
   "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
   unfolding continuous_on_def
   by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
 
 lemma continuous_on_cong_simp:
   "s = t \<Longrightarrow> (\<And>x. x \<in> t =simp=> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
   unfolding simp_implies_def by (rule continuous_on_cong)
 
 lemma continuous_on_topological:
   "continuous_on s f \<longleftrightarrow>
     (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
   unfolding continuous_on_def tendsto_def eventually_at_topological by metis
 
 lemma continuous_on_open_invariant:
   "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
 proof safe
   fix B :: "'b set"
   assume "continuous_on s f" "open B"
   then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
     by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
   then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B"
     unfolding bchoice_iff ..
   then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
     by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
 next
   assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
   show "continuous_on s f"
     unfolding continuous_on_topological
   proof safe
     fix x B
     assume "x \<in> s" "open B" "f x \<in> B"
     with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s"
       by auto
     with \<open>x \<in> s\<close> \<open>f x \<in> B\<close> show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
       by (intro exI[of _ A]) auto
   qed
 qed
 
 lemma continuous_on_open_vimage:
   "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
   unfolding continuous_on_open_invariant
   by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
 
 corollary continuous_imp_open_vimage:
   assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
   shows "open (f -` B)"
   by (metis assms continuous_on_open_vimage le_iff_inf)
 
 corollary open_vimage[continuous_intros]:
   assumes "open s"
     and "continuous_on UNIV f"
   shows "open (f -` s)"
   using assms by (simp add: continuous_on_open_vimage [OF open_UNIV])
 
 lemma continuous_on_closed_invariant:
   "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
 proof -
   have *: "(\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
     for P Q :: "'b set \<Rightarrow> bool"
     by (metis double_compl)
   show ?thesis
     unfolding continuous_on_open_invariant
     by (intro *) (auto simp: open_closed[symmetric])
 qed
 
 lemma continuous_on_closed_vimage:
   "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
   unfolding continuous_on_closed_invariant
   by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
 
 corollary closed_vimage_Int[continuous_intros]:
   assumes "closed s"
     and "continuous_on t f"
     and t: "closed t"
   shows "closed (f -` s \<inter> t)"
   using assms by (simp add: continuous_on_closed_vimage [OF t])
 
 corollary closed_vimage[continuous_intros]:
   assumes "closed s"
     and "continuous_on UNIV f"
   shows "closed (f -` s)"
   using closed_vimage_Int [OF assms] by simp
 
 lemma continuous_on_empty [simp]: "continuous_on {} f"
   by (simp add: continuous_on_def)
 
 lemma continuous_on_sing [simp]: "continuous_on {x} f"
   by (simp add: continuous_on_def at_within_def)
 
 lemma continuous_on_open_Union:
   "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f"
   unfolding continuous_on_def
   by safe (metis open_Union at_within_open UnionI)
 
 lemma continuous_on_open_UN:
   "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow>
     continuous_on (\<Union>s\<in>S. A s) f"
   by (rule continuous_on_open_Union) auto
 
 lemma continuous_on_open_Un:
   "open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
   using continuous_on_open_Union [of "{s,t}"] by auto
 
 lemma continuous_on_closed_Un:
   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
   by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
 
 lemma continuous_on_closed_Union:
   assumes "finite I"
     "\<And>i. i \<in> I \<Longrightarrow> closed (U i)"
     "\<And>i. i \<in> I \<Longrightarrow> continuous_on (U i) f"
   shows "continuous_on (\<Union> i \<in> I. U i) f"
   using assms
   by (induction I) (auto intro!: continuous_on_closed_Un)
 
 lemma continuous_on_If:
   assumes closed: "closed s" "closed t"
     and cont: "continuous_on s f" "continuous_on t g"
     and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x"
   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
     (is "continuous_on _ ?h")
 proof-
   from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
     by auto
   with cont have "continuous_on s ?h" "continuous_on t ?h"
     by simp_all
   with closed show ?thesis
     by (rule continuous_on_closed_Un)
 qed
 
 lemma continuous_on_cases:
   "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
     \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
     continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
   by (rule continuous_on_If) auto
 
 lemma continuous_on_id[continuous_intros,simp]: "continuous_on s (\<lambda>x. x)"
   unfolding continuous_on_def by fast
 
 lemma continuous_on_id'[continuous_intros,simp]: "continuous_on s id"
   unfolding continuous_on_def id_def by fast
 
 lemma continuous_on_const[continuous_intros,simp]: "continuous_on s (\<lambda>x. c)"
   unfolding continuous_on_def by auto
 
 lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
   unfolding continuous_on_def
   by (metis subset_eq tendsto_within_subset)
 
 lemma continuous_on_compose[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g \<circ> f)"
   unfolding continuous_on_topological by simp metis
 
 lemma continuous_on_compose2:
   "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
   using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def)
 
 lemma continuous_on_generate_topology:
   assumes *: "open = generate_topology X"
     and **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
   shows "continuous_on A f"
   unfolding continuous_on_open_invariant
 proof safe
   fix B :: "'a set"
   assume "open B"
   then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
     unfolding *
   proof induct
     case (UN K)
     then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A"
       by metis
     then show ?case
       by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast
   qed (auto intro: **)
 qed
 
 lemma continuous_onI_mono:
   fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order,linorder_topology}"
   assumes "open (f`A)"
     and mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
   shows "continuous_on A f"
 proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
   have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y"
     by (auto simp: not_le[symmetric] mono)
   have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x" if a: "a \<in> A" and fa: "f a < b" for a b
   proof -
     obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
       using open_right[OF \<open>open (f`A)\<close>, of "f a" b] a fa
       by auto
     obtain z where z: "f a < z" "z < min b y"
       using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto
     then obtain c where "z = f c" "c \<in> A"
       using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
     with a z show ?thesis
       by (auto intro!: exI[of _ c] simp: monoD)
   qed
   then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b
     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"])
        (auto intro: le_less_trans[OF mono] less_imp_le)
 
   have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a" if a: "a \<in> A" and fa: "b < f a" for a b
   proof -
     note a fa
     moreover
     obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
       using open_left[OF \<open>open (f`A)\<close>, of "f a" b]  a fa
       by auto
     then obtain z where z: "max b y < z" "z < f a"
       using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto
     then obtain c where "z = f c" "c \<in> A"
       using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
     with a z show ?thesis
       by (auto intro!: exI[of _ c] simp: monoD)
   qed
   then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b
     by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"])
        (auto intro: less_le_trans[OF _ mono] less_imp_le)
 qed
 
 lemma continuous_on_IccI:
   "\<lbrakk>(f \<longlongrightarrow> f a) (at_right a);
     (f \<longlongrightarrow> f b) (at_left b);
     (\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> f \<midarrow>x\<rightarrow> f x); a < b\<rbrakk> \<Longrightarrow>
     continuous_on {a .. b} f"
   for a::"'a::linorder_topology"
   using at_within_open[of _ "{a<..<b}"]
   by (auto simp: continuous_on_def at_within_Icc_at_right at_within_Icc_at_left le_less
       at_within_Icc_at)
 
 lemma
   fixes a b::"'a::linorder_topology"
   assumes "continuous_on {a .. b} f" "a < b"
   shows continuous_on_Icc_at_rightD: "(f \<longlongrightarrow> f a) (at_right a)"
     and continuous_on_Icc_at_leftD: "(f \<longlongrightarrow> f b) (at_left b)"
   using assms
   by (auto simp: at_within_Icc_at_right at_within_Icc_at_left continuous_on_def
       dest: bspec[where x=a] bspec[where x=b])
 
 lemma continuous_on_discrete [simp]:
   "continuous_on A (f :: 'a :: discrete_topology \<Rightarrow> _)"
   by (auto simp: continuous_on_def at_discrete)
 
 subsubsection \<open>Continuity at a point\<close>
 
 definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
   where "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
 
 lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
   unfolding continuous_def by auto
 
 lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
   by simp
 
 lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> f x) (at x within s)"
   by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
 
 lemma continuous_within_topological:
   "continuous (at x within s) f \<longleftrightarrow>
     (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
   unfolding continuous_within tendsto_def eventually_at_topological by metis
 
 lemma continuous_within_compose[continuous_intros]:
   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
     continuous (at x within s) (g \<circ> f)"
   by (simp add: continuous_within_topological) metis
 
 lemma continuous_within_compose2:
   "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
     continuous (at x within s) (\<lambda>x. g (f x))"
   using continuous_within_compose[of x s f g] by (simp add: comp_def)
 
 lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
   using continuous_within[of x UNIV f] by simp
 
 lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
   unfolding continuous_within by (rule tendsto_ident_at)
 
 lemma continuous_id[continuous_intros, simp]: "continuous (at x within S) id"
   by (simp add: id_def)
 
 lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
   unfolding continuous_def by (rule tendsto_const)
 
 lemma continuous_on_eq_continuous_within:
   "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
   unfolding continuous_on_def continuous_within ..
 
 lemma continuous_discrete [simp]:
   "continuous (at x within A) (f :: 'a :: discrete_topology \<Rightarrow> _)"
   by (auto simp: continuous_def at_discrete)
 
 abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool"
   where "isCont f a \<equiv> continuous (at a) f"
 
 lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a"
   by (rule continuous_at)
 
 lemma isContD: "isCont f x \<Longrightarrow> f \<midarrow>x\<rightarrow> f x"
   by (simp add: isCont_def)
 
 lemma isCont_cong:
   assumes "eventually (\<lambda>x. f x = g x) (nhds x)"
   shows "isCont f x \<longleftrightarrow> isCont g x"
 proof -
   from assms have [simp]: "f x = g x"
     by (rule eventually_nhds_x_imp_x)
   from assms have "eventually (\<lambda>x. f x = g x) (at x)"
     by (auto simp: eventually_at_filter elim!: eventually_mono)
   with assms have "isCont f x \<longleftrightarrow> isCont g x" unfolding isCont_def
     by (intro filterlim_cong) (auto elim!: eventually_mono)
   with assms show ?thesis by simp
 qed
 
 lemma continuous_at_imp_continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
   by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
 
 lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
   by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
 
 lemma continuous_within_open: "a \<in> A \<Longrightarrow> open A \<Longrightarrow> continuous (at a within A) f \<longleftrightarrow> isCont f a"
   by (simp add: at_within_open_NO_MATCH)
 
 lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
   by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_continuous_within)
 
 lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
   unfolding isCont_def by (rule tendsto_compose)
 
 lemma continuous_at_compose[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
   unfolding o_def by (rule isCont_o2)
 
 lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
   unfolding isCont_def by (rule tendsto_compose)
 
 lemma continuous_on_tendsto_compose:
   assumes f_cont: "continuous_on s f"
     and g: "(g \<longlongrightarrow> l) F"
     and l: "l \<in> s"
     and ev: "\<forall>\<^sub>Fx in F. g x \<in> s"
   shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
 proof -
   from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)"
     by (simp add: continuous_on_def)
   have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F"
     by (rule filterlim_If)
        (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g]
              simp: filterlim_at eventually_inf_principal eventually_mono[OF ev])
   show ?thesis
     by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto
 qed
 
 lemma continuous_within_compose3:
   "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
   using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast
 
 lemma filtermap_nhds_open_map:
   assumes cont: "isCont f a"
     and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
   shows "filtermap f (nhds a) = nhds (f a)"
   unfolding filter_eq_iff
 proof safe
   fix P
   assume "eventually P (filtermap f (nhds a))"
   then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. P (f x)"
     by (auto simp: eventually_filtermap eventually_nhds)
   then show "eventually P (nhds (f a))"
     unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
 qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
 
 lemma continuous_at_split:
   "continuous (at x) f \<longleftrightarrow> continuous (at_left x) f \<and> continuous (at_right x) f"
   for x :: "'a::linorder_topology"
   by (simp add: continuous_within filterlim_at_split)
 
 lemma continuous_on_max [continuous_intros]:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. max (f x) (g x))"
   by (auto simp: continuous_on_def intro!: tendsto_max)
 
 lemma continuous_on_min [continuous_intros]:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   shows "continuous_on A f \<Longrightarrow> continuous_on A g \<Longrightarrow> continuous_on A (\<lambda>x. min (f x) (g x))"
   by (auto simp: continuous_on_def intro!: tendsto_min)
 
 lemma continuous_max [continuous_intros]:
   fixes f :: "'a::t2_space \<Rightarrow> 'b::linorder_topology"
   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (max (f x) (g x)))"
   by (simp add: tendsto_max continuous_def)
 
 lemma continuous_min [continuous_intros]:
   fixes f :: "'a::t2_space \<Rightarrow> 'b::linorder_topology"
   shows "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (min (f x) (g x)))"
   by (simp add: tendsto_min continuous_def)
 
 text \<open>
   The following open/closed Collect lemmas are ported from
   Sébastien Gouëzel's \<open>Ergodic_Theory\<close>.
 \<close>
 lemma open_Collect_neq:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "open {x. f x \<noteq> g x}"
 proof (rule openI)
   fix t
   assume "t \<in> {x. f x \<noteq> g x}"
   then obtain U V where *: "open U" "open V" "f t \<in> U" "g t \<in> V" "U \<inter> V = {}"
     by (auto simp add: separation_t2)
   with open_vimage[OF \<open>open U\<close> f] open_vimage[OF \<open>open V\<close> g]
   show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x \<noteq> g x}"
     by (intro exI[of _ "f -` U \<inter> g -` V"]) auto
 qed
 
 lemma closed_Collect_eq:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "closed {x. f x = g x}"
   using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq)
 
 lemma open_Collect_less:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
   shows "open {x. f x < g x}"
 proof (rule openI)
   fix t
   assume t: "t \<in> {x. f x < g x}"
   show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x. f x < g x}"
   proof (cases "\<exists>z. f t < z \<and> z < g t")
     case True
     then obtain z where "f t < z \<and> z < g t" by blast
     then show ?thesis
       using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"]
       by (intro exI[of _ "f -` {..<z} \<inter> g -` {z<..}"]) auto
   next
     case False
     then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}"
       using t by (auto intro: leI)
     show ?thesis
       using open_vimage[OF _ f, of "{..< g t}"] open_vimage[OF _ g, of "{f t <..}"] t
       apply (intro exI[of _ "f -` {..< g t} \<inter> g -` {f t<..}"])
       apply (simp add: open_Int)
       apply (auto simp add: *)
       done
   qed
 qed
 
 lemma closed_Collect_le:
   fixes f g :: "'a :: topological_space \<Rightarrow> 'b::linorder_topology"
   assumes f: "continuous_on UNIV f"
     and g: "continuous_on UNIV g"
   shows "closed {x. f x \<le> g x}"
   using open_Collect_less [OF g f]
   by (simp add: closed_def Collect_neg_eq[symmetric] not_le)
 
 
 subsubsection \<open>Open-cover compactness\<close>
 
 context topological_space
 begin
 
 definition compact :: "'a set \<Rightarrow> bool" where
 compact_eq_Heine_Borel:  (* This name is used for backwards compatibility *)
     "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
 
 lemma compactI:
   assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union>C'"
   shows "compact s"
   unfolding compact_eq_Heine_Borel using assms by metis
 
 lemma compact_empty[simp]: "compact {}"
   by (auto intro!: compactI)
 
 lemma compactE: (*related to COMPACT_IMP_HEINE_BOREL in HOL Light*)
   assumes "compact S" "S \<subseteq> \<Union>\<T>" "\<And>B. B \<in> \<T> \<Longrightarrow> open B"
   obtains \<T>' where "\<T>' \<subseteq> \<T>" "finite \<T>'" "S \<subseteq> \<Union>\<T>'"
   by (meson assms compact_eq_Heine_Borel)
 
 lemma compactE_image:
   assumes "compact S"
     and opn: "\<And>T. T \<in> C \<Longrightarrow> open (f T)"
     and S: "S \<subseteq> (\<Union>c\<in>C. f c)"
   obtains C' where "C' \<subseteq> C" and "finite C'" and "S \<subseteq> (\<Union>c\<in>C'. f c)"
     apply (rule compactE[OF \<open>compact S\<close> S])
     using opn apply force
     by (metis finite_subset_image)
 
 lemma compact_Int_closed [intro]:
   assumes "compact S"
     and "closed T"
   shows "compact (S \<inter> T)"
 proof (rule compactI)
   fix C
   assume C: "\<forall>c\<in>C. open c"
   assume cover: "S \<inter> T \<subseteq> \<Union>C"
   from C \<open>closed T\<close> have "\<forall>c\<in>C \<union> {- T}. open c"
     by auto
   moreover from cover have "S \<subseteq> \<Union>(C \<union> {- T})"
     by auto
   ultimately have "\<exists>D\<subseteq>C \<union> {- T}. finite D \<and> S \<subseteq> \<Union>D"
     using \<open>compact S\<close> unfolding compact_eq_Heine_Borel by auto
   then obtain D where "D \<subseteq> C \<union> {- T} \<and> finite D \<and> S \<subseteq> \<Union>D" ..
   then show "\<exists>D\<subseteq>C. finite D \<and> S \<inter> T \<subseteq> \<Union>D"
     by (intro exI[of _ "D - {-T}"]) auto
 qed
 
 lemma compact_diff: "\<lbrakk>compact S; open T\<rbrakk> \<Longrightarrow> compact(S - T)"
   by (simp add: Diff_eq compact_Int_closed open_closed)
 
 lemma inj_setminus: "inj_on uminus (A::'a set set)"
   by (auto simp: inj_on_def)
 
 
 subsection \<open>Finite intersection property\<close>
 
 lemma compact_fip:
   "compact U \<longleftrightarrow>
     (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
   (is "_ \<longleftrightarrow> ?R")
 proof (safe intro!: compact_eq_Heine_Borel[THEN iffD2])
   fix A
   assume "compact U"
   assume A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
   assume fin: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
   from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
     by auto
   with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
     unfolding compact_eq_Heine_Borel by (metis subset_image_iff)
   with fin[THEN spec, of B] show False
     by (auto dest: finite_imageD intro: inj_setminus)
 next
   fix A
   assume ?R
   assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
   then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
     by auto
   with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
     by (metis subset_image_iff)
   then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
     by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
 qed
 
 lemma compact_imp_fip:
   assumes "compact S"
     and "\<And>T. T \<in> F \<Longrightarrow> closed T"
     and "\<And>F'. finite F' \<Longrightarrow> F' \<subseteq> F \<Longrightarrow> S \<inter> (\<Inter>F') \<noteq> {}"
   shows "S \<inter> (\<Inter>F) \<noteq> {}"
   using assms unfolding compact_fip by auto
 
 lemma compact_imp_fip_image:
   assumes "compact s"
     and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
     and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
   shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
 proof -
   note \<open>compact s\<close>
   moreover from P have "\<forall>i \<in> f ` I. closed i"
     by blast
   moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
     apply rule
     apply rule
     apply (erule conjE)
   proof -
     fix A :: "'a set set"
     assume "finite A" and "A \<subseteq> f ` I"
     then obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
       using finite_subset_image [of A f I] by blast
     with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}"
       by simp
   qed
   ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}"
     by (metis compact_imp_fip)
   then show ?thesis by simp
 qed
 
 end
 
 lemma (in t2_space) compact_imp_closed:
   assumes "compact s"
   shows "closed s"
   unfolding closed_def
 proof (rule openI)
   fix y
   assume "y \<in> - s"
   let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
   have "s \<subseteq> \<Union>?C"
   proof
     fix x
     assume "x \<in> s"
     with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp
     then have "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
       by (rule hausdorff)
     with \<open>x \<in> s\<close> show "x \<in> \<Union>?C"
       unfolding eventually_nhds by auto
   qed
   then obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
     by (rule compactE [OF \<open>compact s\<close>]) auto
   from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)"
     by auto
   with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
     by (simp add: eventually_ball_finite)
   with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
     by (auto elim!: eventually_mono)
   then show "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
     by (simp add: eventually_nhds subset_eq)
 qed
 
 lemma compact_continuous_image:
   assumes f: "continuous_on s f"
     and s: "compact s"
   shows "compact (f ` s)"
 proof (rule compactI)
   fix C
   assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
   with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
     unfolding continuous_on_open_invariant by blast
   then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
     unfolding bchoice_iff ..
   with cover have "\<And>c. c \<in> C \<Longrightarrow> open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
     by (fastforce simp add: subset_eq set_eq_iff)+
   from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
   with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
     by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
 qed
 
 lemma continuous_on_inv:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes "continuous_on s f"
     and "compact s"
     and "\<forall>x\<in>s. g (f x) = x"
   shows "continuous_on (f ` s) g"
   unfolding continuous_on_topological
 proof (clarsimp simp add: assms(3))
   fix x :: 'a and B :: "'a set"
   assume "x \<in> s" and "open B" and "x \<in> B"
   have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
     using assms(3) by (auto, metis)
   have "continuous_on (s - B) f"
     using \<open>continuous_on s f\<close> Diff_subset
     by (rule continuous_on_subset)
   moreover have "compact (s - B)"
     using \<open>open B\<close> and \<open>compact s\<close>
     unfolding Diff_eq by (intro compact_Int_closed closed_Compl)
   ultimately have "compact (f ` (s - B))"
     by (rule compact_continuous_image)
   then have "closed (f ` (s - B))"
     by (rule compact_imp_closed)
   then have "open (- f ` (s - B))"
     by (rule open_Compl)
   moreover have "f x \<in> - f ` (s - B)"
     using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1)
   moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
     by (simp add: 1)
   ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
     by fast
 qed
 
 lemma continuous_on_inv_into:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes s: "continuous_on s f" "compact s"
     and f: "inj_on f s"
   shows "continuous_on (f ` s) (the_inv_into s f)"
   by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
 
 lemma (in linorder_topology) compact_attains_sup:
   assumes "compact S" "S \<noteq> {}"
   shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
 proof (rule classical)
   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
     by (metis not_le)
   then have "\<And>s. s\<in>S \<Longrightarrow> open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
     by auto
   with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
     by (metis compactE_image)
   with \<open>S \<noteq> {}\<close> have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
     by (auto intro!: Max_in)
   with C have "S \<subseteq> {..< Max (t`C)}"
     by (auto intro: less_le_trans simp: subset_eq)
   with t Max \<open>C \<subseteq> S\<close> show ?thesis
     by fastforce
 qed
 
 lemma (in linorder_topology) compact_attains_inf:
   assumes "compact S" "S \<noteq> {}"
   shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
 proof (rule classical)
   assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
   then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
     by (metis not_le)
   then have "\<And>s. s\<in>S \<Longrightarrow> open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
     by auto
   with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
     by (metis compactE_image)
   with \<open>S \<noteq> {}\<close> have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
     by (auto intro!: Min_in)
   with C have "S \<subseteq> {Min (t`C) <..}"
     by (auto intro: le_less_trans simp: subset_eq)
   with t Min \<open>C \<subseteq> S\<close> show ?thesis
     by fastforce
 qed
 
 lemma continuous_attains_sup:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s.  f y \<le> f x)"
   using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
 
 lemma continuous_attains_inf:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
   shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"
   using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
 
 
 subsection \<open>Connectedness\<close>
 
 context topological_space
 begin
 
 definition "connected S \<longleftrightarrow>
   \<not> (\<exists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
 
 lemma connectedI:
   "(\<And>A B. open A \<Longrightarrow> open B \<Longrightarrow> A \<inter> U \<noteq> {} \<Longrightarrow> B \<inter> U \<noteq> {} \<Longrightarrow> A \<inter> B \<inter> U = {} \<Longrightarrow> U \<subseteq> A \<union> B \<Longrightarrow> False)
   \<Longrightarrow> connected U"
   by (auto simp: connected_def)
 
 lemma connected_empty [simp]: "connected {}"
   by (auto intro!: connectedI)
 
 lemma connected_sing [simp]: "connected {x}"
   by (auto intro!: connectedI)
 
 lemma connectedD:
   "connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}"
   by (auto simp: connected_def)
 
 end
 
 lemma connected_closed:
   "connected s \<longleftrightarrow>
     \<not> (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
   apply (simp add: connected_def del: ex_simps, safe)
    apply (drule_tac x="-A" in spec)
    apply (drule_tac x="-B" in spec)
    apply (fastforce simp add: closed_def [symmetric])
   apply (drule_tac x="-A" in spec)
   apply (drule_tac x="-B" in spec)
   apply (fastforce simp add: open_closed [symmetric])
   done
 
 lemma connected_closedD:
   "\<lbrakk>connected s; A \<inter> B \<inter> s = {}; s \<subseteq> A \<union> B; closed A; closed B\<rbrakk> \<Longrightarrow> A \<inter> s = {} \<or> B \<inter> s = {}"
   by (simp add: connected_closed)
 
 lemma connected_Union:
   assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s"
     and ne: "\<Inter>S \<noteq> {}"
   shows "connected(\<Union>S)"
 proof (rule connectedI)
   fix A B
   assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}"
     and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
   have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}"
     using disj by auto
   obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}"
     using Alap by auto
   obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}"
     using Blap by auto
   obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s"
     using ne by auto
   then have "x \<in> \<Union>S"
     using \<open>sa \<in> S\<close> by blast
   then have "x \<in> A \<or> x \<in> B"
     using cover by auto
   then show False
     using cs [unfolded connected_def]
     by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
 qed
 
 lemma connected_Un: "connected s \<Longrightarrow> connected t \<Longrightarrow> s \<inter> t \<noteq> {} \<Longrightarrow> connected (s \<union> t)"
   using connected_Union [of "{s,t}"] by auto
 
 lemma connected_diff_open_from_closed:
   assumes st: "s \<subseteq> t"
     and tu: "t \<subseteq> u"
     and s: "open s"
     and t: "closed t"
     and u: "connected u"
     and ts: "connected (t - s)"
   shows "connected(u - s)"
 proof (rule connectedI)
   fix A B
   assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}"
     and disj: "A \<inter> B \<inter> (u - s) = {}"
     and cover: "u - s \<subseteq> A \<union> B"
   then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}"
     using st ts tu connectedD [of "t-s" "A" "B"] by auto
   then show False
   proof cases
     case 1
     then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}"
       using disj st by auto
     moreover have "u \<subseteq> (A - t) \<union> (B \<union> s)"
       using 1 cover by auto
     ultimately show False
       using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u by auto
   next
     case 2
     then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}"
       using disj st by auto
     moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)"
       using 2 cover by auto
     ultimately show False
       using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u by auto
   qed
 qed
 
 lemma connected_iff_const:
   fixes S :: "'a::topological_space set"
   shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
 proof safe
   fix P :: "'a \<Rightarrow> bool"
   assume "connected S" "continuous_on S P"
   then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S"
     unfolding continuous_on_open_invariant by (simp add: open_discrete)
   from this[of True] this[of False]
   obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S"
     by meson
   then have "t \<inter> S = {} \<or> f \<inter> S = {}"
     by (intro connectedD[OF \<open>connected S\<close>])  auto
   then show "\<exists>c. \<forall>s\<in>S. P s = c"
   proof (rule disjE)
     assume "t \<inter> S = {}"
     then show ?thesis
       unfolding * by (intro exI[of _ False]) auto
   next
     assume "f \<inter> S = {}"
     then show ?thesis
       unfolding * by (intro exI[of _ True]) auto
   qed
 next
   assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)"
   show "connected S"
   proof (rule connectedI)
     fix A B
     assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
     have "continuous_on S (\<lambda>x. x \<in> A)"
       unfolding continuous_on_open_invariant
     proof safe
       fix C :: "bool set"
       have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}"
         using subset_UNIV[of C] unfolding UNIV_bool by auto
       with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S"
         by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto
     qed
     from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c"
       by blast
     with * show False
       by (cases c) auto
   qed
 qed
 
 lemma connectedD_const: "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
   for P :: "'a::topological_space \<Rightarrow> bool"
   by (auto simp: connected_iff_const)
 
 lemma connectedI_const:
   "(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S"
   by (auto simp: connected_iff_const)
 
 lemma connected_local_const:
   assumes "connected A" "a \<in> A" "b \<in> A"
     and *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
   shows "f a = f b"
 proof -
   obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
     "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
     using * unfolding eventually_at_topological by metis
   let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b"
   have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
     using \<open>connected A\<close> S \<open>a\<in>A\<close>
     by (intro connectedD) (auto, metis)
   then show "f a = f b"
   proof
     assume "?N \<inter> A = {}"
     then have "\<forall>x\<in>A. f a = f x"
       using S(1) by auto
     with \<open>b\<in>A\<close> show ?thesis by auto
   next
     assume "?P \<inter> A = {}" then show ?thesis
       using \<open>a \<in> A\<close> S(1)[of a] by auto
   qed
 qed
 
 lemma (in linorder_topology) connectedD_interval:
   assumes "connected U"
     and xy: "x \<in> U" "y \<in> U"
     and "x \<le> z" "z \<le> y"
   shows "z \<in> U"
 proof -
   have eq: "{..<z} \<union> {z<..} = - {z}"
     by auto
   have "\<not> connected U" if "z \<notin> U" "x < z" "z < y"
     using xy that
     apply (simp only: connected_def simp_thms)
     apply (rule_tac exI[of _ "{..< z}"])
     apply (rule_tac exI[of _ "{z <..}"])
     apply (auto simp add: eq)
     done
   with assms show "z \<in> U"
     by (metis less_le)
 qed
 
 lemma (in linorder_topology) not_in_connected_cases:
   assumes conn: "connected S"
   assumes nbdd: "x \<notin> S"
   assumes ne: "S \<noteq> {}"
   obtains "bdd_above S" "\<And>y. y \<in> S \<Longrightarrow> x \<ge> y" | "bdd_below S" "\<And>y. y \<in> S \<Longrightarrow> x \<le> y"
 proof -
   obtain s where "s \<in> S" using ne by blast
   {
     assume "s \<le> x"
     have "False" if "x \<le> y" "y \<in> S" for y
       using connectedD_interval[OF conn \<open>s \<in> S\<close> \<open>y \<in> S\<close> \<open>s \<le> x\<close> \<open>x \<le> y\<close>] \<open>x \<notin> S\<close>
       by simp
     then have wit: "y \<in> S \<Longrightarrow> x \<ge> y" for y
       using le_cases by blast
     then have "bdd_above S"
       by (rule local.bdd_aboveI)
     note this wit
   } moreover {
     assume "x \<le> s"
     have "False" if "x \<ge> y" "y \<in> S" for y
       using connectedD_interval[OF conn \<open>y \<in> S\<close> \<open>s \<in> S\<close> \<open>x \<ge> y\<close> \<open>s \<ge> x\<close> ] \<open>x \<notin> S\<close>
       by simp
     then have wit: "y \<in> S \<Longrightarrow> x \<le> y" for y
       using le_cases by blast
     then have "bdd_below S"
       by (rule bdd_belowI)
     note this wit
   } ultimately show ?thesis
     by (meson le_cases that)
 qed
 
 lemma connected_continuous_image:
   assumes *: "continuous_on s f"
     and "connected s"
   shows "connected (f ` s)"
 proof (rule connectedI_const)
   fix P :: "'b \<Rightarrow> bool"
   assume "continuous_on (f ` s) P"
   then have "continuous_on s (P \<circ> f)"
     by (rule continuous_on_compose[OF *])
   from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c"
     by auto
 qed
 
 
 section \<open>Linear Continuum Topologies\<close>
 
 class linear_continuum_topology = linorder_topology + linear_continuum
 begin
 
 lemma Inf_notin_open:
   assumes A: "open A"
     and bnd: "\<forall>a\<in>A. x < a"
   shows "Inf A \<notin> A"
 proof
   assume "Inf A \<in> A"
   then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
     using open_left[of A "Inf A" x] assms by auto
   with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
     by (auto simp: subset_eq)
   then show False
     using cInf_lower[OF \<open>c \<in> A\<close>] bnd
     by (metis not_le less_imp_le bdd_belowI)
 qed
 
 lemma Sup_notin_open:
   assumes A: "open A"
     and bnd: "\<forall>a\<in>A. a < x"
   shows "Sup A \<notin> A"
 proof
   assume "Sup A \<in> A"
   with assms obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
     using open_right[of A "Sup A" x] by auto
   with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
     by (auto simp: subset_eq)
   then show False
     using cSup_upper[OF \<open>c \<in> A\<close>] bnd
     by (metis less_imp_le not_le bdd_aboveI)
 qed
 
 end
 
 instance linear_continuum_topology \<subseteq> perfect_space
 proof
   fix x :: 'a
   obtain y where "x < y \<or> y < x"
     using ex_gt_or_lt [of x] ..
   with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "\<not> open {x}"
     by auto
 qed
 
 lemma connectedI_interval:
   fixes U :: "'a :: linear_continuum_topology set"
   assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
   shows "connected U"
 proof (rule connectedI)
   {
     fix A B
     assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
     fix x y
     assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
 
     let ?z = "Inf (B \<inter> {x <..})"
 
     have "x \<le> ?z" "?z \<le> y"
       using \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest)
     with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U"
       by (rule *)
     moreover have "?z \<notin> B \<inter> {x <..}"
       using \<open>open B\<close> by (intro Inf_notin_open) auto
     ultimately have "?z \<in> A"
       using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto
     have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" if "?z < y"
     proof -
       obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
         using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto
       moreover obtain b where "b \<in> B" "x < b" "b < min a y"
         using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] \<open>?z < a\<close> \<open>?z < y\<close> \<open>x < y\<close> \<open>y \<in> B\<close>
         by auto
       moreover have "?z \<le> b"
         using \<open>b \<in> B\<close> \<open>x < b\<close>
         by (intro cInf_lower) auto
       moreover have "b \<in> U"
         using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close>
         by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le)
       ultimately show ?thesis
         by (intro bexI[of _ b]) auto
     qed
     then have False
       using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close>
       unfolding le_less by blast
   }
   note not_disjoint = this
 
   fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
   moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
   moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
   moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
   ultimately show False
     by (cases x y rule: linorder_cases) auto
 qed
 
 lemma connected_iff_interval: "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
   for U :: "'a::linear_continuum_topology set"
   by (auto intro: connectedI_interval dest: connectedD_interval)
 
 lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
   by (simp add: connected_iff_interval)
 
 lemma connected_Ioi[simp]: "connected {a<..}"
   for a :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Ici[simp]: "connected {a..}"
   for a :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Iio[simp]: "connected {..<a}"
   for a :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Iic[simp]: "connected {..a}"
   for a :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Ioo[simp]: "connected {a<..<b}"
   for a b :: "'a::linear_continuum_topology"
   unfolding connected_iff_interval by auto
 
 lemma connected_Ioc[simp]: "connected {a<..b}"
   for a b :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Ico[simp]: "connected {a..<b}"
   for a b :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_Icc[simp]: "connected {a..b}"
   for a b :: "'a::linear_continuum_topology"
   by (auto simp: connected_iff_interval)
 
 lemma connected_contains_Ioo:
   fixes A :: "'a :: linorder_topology set"
   assumes "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
   using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le)
 
 lemma connected_contains_Icc:
   fixes A :: "'a::linorder_topology set"
   assumes "connected A" "a \<in> A" "b \<in> A"
   shows "{a..b} \<subseteq> A"
 proof
   fix x assume "x \<in> {a..b}"
   then have "x = a \<or> x = b \<or> x \<in> {a<..<b}"
     by auto
   then show "x \<in> A"
     using assms connected_contains_Ioo[of A a b] by auto
 qed
 
 
 subsection \<open>Intermediate Value Theorem\<close>
 
 lemma IVT':
   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
     and *: "continuous_on {a .. b} f"
   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
 proof -
   have "connected {a..b}"
     unfolding connected_iff_interval by auto
   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
   show ?thesis
     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
 qed
 
 lemma IVT2':
   fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
   assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
     and *: "continuous_on {a .. b} f"
   shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
 proof -
   have "connected {a..b}"
     unfolding connected_iff_interval by auto
   from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
   show ?thesis
     by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
 qed
 
 lemma IVT:
   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow>
     \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
   by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
 
 lemma IVT2:
   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow>
     \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
   by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
 
 lemma continuous_inj_imp_mono:
   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   assumes x: "a < x" "x < b"
     and cont: "continuous_on {a..b} f"
     and inj: "inj_on f {a..b}"
   shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
 proof -
   note I = inj_on_eq_iff[OF inj]
   {
     assume "f x < f a" "f x < f b"
     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"
       using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
     with x I have False by auto
   }
   moreover
   {
     assume "f a < f x" "f b < f x"
     then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"
       using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
       by (auto simp: continuous_on_subset[OF cont] less_imp_le)
     with x I have False by auto
   }
   ultimately show ?thesis
     using I[of a x] I[of x b] x less_trans[OF x]
     by (auto simp add: le_less less_imp_neq neq_iff)
 qed
 
 lemma continuous_at_Sup_mono:
   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
     'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "mono f"
     and cont: "continuous (at_left (Sup S)) f"
     and S: "S \<noteq> {}" "bdd_above S"
   shows "f (Sup S) = (SUP s\<in>S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
     using cont unfolding continuous_within .
   show "f (Sup S) \<le> (SUP s\<in>S. f s)"
   proof cases
     assume "Sup S \<in> S"
     then show ?thesis
       by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>)
   next
     assume "Sup S \<notin> S"
     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
       by auto
     with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
       unfolding less_le by (blast intro: cSup_upper)
     show ?thesis
     proof (rule ccontr)
       assume "\<not> ?thesis"
       with order_tendstoD(1)[OF f, of "SUP s\<in>S. f s"] obtain b where "b < Sup S"
         and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s\<in>S. f s) < f y"
         by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
         using less_cSupD[of S b] by auto
       with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
         unfolding less_le by (blast intro: cSup_upper)
       from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_mono[of f]]
       show False
         by (auto simp: assms)
     qed
   qed
 qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S)
 
 lemma continuous_at_Sup_antimono:
   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
     'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "antimono f"
     and cont: "continuous (at_left (Sup S)) f"
     and S: "S \<noteq> {}" "bdd_above S"
   shows "f (Sup S) = (INF s\<in>S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
     using cont unfolding continuous_within .
   show "(INF s\<in>S. f s) \<le> f (Sup S)"
   proof cases
     assume "Sup S \<in> S"
     then show ?thesis
       by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>)
   next
     assume "Sup S \<notin> S"
     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
       by auto
     with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
       unfolding less_le by (blast intro: cSup_upper)
     show ?thesis
     proof (rule ccontr)
       assume "\<not> ?thesis"
       with order_tendstoD(2)[OF f, of "INF s\<in>S. f s"] obtain b where "b < Sup S"
         and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> f y < (INF s\<in>S. f s)"
         by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
         using less_cSupD[of S b] by auto
       with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
         unfolding less_le by (blast intro: cSup_upper)
       from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cINF_lower[OF bdd_below_image_antimono, of f S c] \<open>c \<in> S\<close>
       show False
         by (auto simp: assms)
     qed
   qed
 qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S)
 
 lemma continuous_at_Inf_mono:
   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
     'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "mono f"
     and cont: "continuous (at_right (Inf S)) f"
     and S: "S \<noteq> {}" "bdd_below S"
   shows "f (Inf S) = (INF s\<in>S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
     using cont unfolding continuous_within .
   show "(INF s\<in>S. f s) \<le> f (Inf S)"
   proof cases
     assume "Inf S \<in> S"
     then show ?thesis
       by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>)
   next
     assume "Inf S \<notin> S"
     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
       by auto
     with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
       unfolding less_le by (blast intro: cInf_lower)
     show ?thesis
     proof (rule ccontr)
       assume "\<not> ?thesis"
       with order_tendstoD(2)[OF f, of "INF s\<in>S. f s"] obtain b where "Inf S < b"
         and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> f y < (INF s\<in>S. f s)"
         by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
         using cInf_lessD[of S b] by auto
       with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
         unfolding less_le by (blast intro: cInf_lower)
       from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cINF_lower[OF bdd_below_image_mono[of f] \<open>c \<in> S\<close>]
       show False
         by (auto simp: assms)
     qed
   qed
 qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>)
 
 lemma continuous_at_Inf_antimono:
   fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} \<Rightarrow>
     'b::{linorder_topology,conditionally_complete_linorder}"
   assumes "antimono f"
     and cont: "continuous (at_right (Inf S)) f"
     and S: "S \<noteq> {}" "bdd_below S"
   shows "f (Inf S) = (SUP s\<in>S. f s)"
 proof (rule antisym)
   have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))"
     using cont unfolding continuous_within .
   show "f (Inf S) \<le> (SUP s\<in>S. f s)"
   proof cases
     assume "Inf S \<in> S"
     then show ?thesis
       by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>)
   next
     assume "Inf S \<notin> S"
     from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
       by auto
     with \<open>Inf S \<notin> S\<close> S have "Inf S < s"
       unfolding less_le by (blast intro: cInf_lower)
     show ?thesis
     proof (rule ccontr)
       assume "\<not> ?thesis"
       with order_tendstoD(1)[OF f, of "SUP s\<in>S. f s"] obtain b where "Inf S < b"
         and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> (SUP s\<in>S. f s) < f y"
         by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>])
       with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b"
         using cInf_lessD[of S b] by auto
       with \<open>Inf S \<notin> S\<close> S have "Inf S < c"
         unfolding less_le by (blast intro: cInf_lower)
       from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_antimono[of f]]
       show False
         by (auto simp: assms)
     qed
   qed
 qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S)
 
 
 subsection \<open>Uniform spaces\<close>
 
 class uniformity =
   fixes uniformity :: "('a \<times> 'a) filter"
 begin
 
 abbreviation uniformity_on :: "'a set \<Rightarrow> ('a \<times> 'a) filter"
   where "uniformity_on s \<equiv> inf uniformity (principal (s\<times>s))"
 
 end
 
 lemma uniformity_Abort:
   "uniformity =
     Filter.abstract_filter (\<lambda>u. Code.abort (STR ''uniformity is not executable'') (\<lambda>u. uniformity))"
   by simp
 
 class open_uniformity = "open" + uniformity +
   assumes open_uniformity:
     "\<And>U. open U \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
 begin
 
 subclass topological_space
   by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+
 
 end
 
 class uniform_space = open_uniformity +
   assumes uniformity_refl: "eventually E uniformity \<Longrightarrow> E (x, x)"
     and uniformity_sym: "eventually E uniformity \<Longrightarrow> eventually (\<lambda>(x, y). E (y, x)) uniformity"
     and uniformity_trans:
       "eventually E uniformity \<Longrightarrow>
         \<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
 begin
 
 lemma uniformity_bot: "uniformity \<noteq> bot"
   using uniformity_refl by auto
 
 lemma uniformity_trans':
   "eventually E uniformity \<Longrightarrow>
     eventually (\<lambda>((x, y), (y', z)). y = y' \<longrightarrow> E (x, z)) (uniformity \<times>\<^sub>F uniformity)"
   by (drule uniformity_trans) (auto simp add: eventually_prod_same)
 
 lemma uniformity_transE:
   assumes "eventually E uniformity"
   obtains D where "eventually D uniformity" "\<And>x y z. D (x, y) \<Longrightarrow> D (y, z) \<Longrightarrow> E (x, z)"
   using uniformity_trans [OF assms] by auto
 
 lemma eventually_nhds_uniformity:
   "eventually P (nhds x) \<longleftrightarrow> eventually (\<lambda>(x', y). x' = x \<longrightarrow> P y) uniformity"
   (is "_ \<longleftrightarrow> ?N P x")
   unfolding eventually_nhds
 proof safe
   assume *: "?N P x"
   have "?N (?N P) x" if "?N P x" for x
   proof -
     from that obtain D where ev: "eventually D uniformity"
       and D: "D (a, b) \<Longrightarrow> D (b, c) \<Longrightarrow> case (a, c) of (x', y) \<Rightarrow> x' = x \<longrightarrow> P y" for a b c
       by (rule uniformity_transE) simp
     from ev show ?thesis
       by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split)
   qed
   then have "open {x. ?N P x}"
     by (simp add: open_uniformity)
   then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>x\<in>S. P x)"
     by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *)
 qed (force simp add: open_uniformity elim: eventually_mono)
 
 
 subsubsection \<open>Totally bounded sets\<close>
 
 definition totally_bounded :: "'a set \<Rightarrow> bool"
   where "totally_bounded S \<longleftrightarrow>
     (\<forall>E. eventually E uniformity \<longrightarrow> (\<exists>X. finite X \<and> (\<forall>s\<in>S. \<exists>x\<in>X. E (x, s))))"
 
 lemma totally_bounded_empty[iff]: "totally_bounded {}"
   by (auto simp add: totally_bounded_def)
 
 lemma totally_bounded_subset: "totally_bounded S \<Longrightarrow> T \<subseteq> S \<Longrightarrow> totally_bounded T"
   by (fastforce simp add: totally_bounded_def)
 
 lemma totally_bounded_Union[intro]:
   assumes M: "finite M" "\<And>S. S \<in> M \<Longrightarrow> totally_bounded S"
   shows "totally_bounded (\<Union>M)"
   unfolding totally_bounded_def
 proof safe
   fix E
   assume "eventually E uniformity"
   with M obtain X where "\<forall>S\<in>M. finite (X S) \<and> (\<forall>s\<in>S. \<exists>x\<in>X S. E (x, s))"
     by (metis totally_bounded_def)
   with \<open>finite M\<close> show "\<exists>X. finite X \<and> (\<forall>s\<in>\<Union>M. \<exists>x\<in>X. E (x, s))"
     by (intro exI[of _ "\<Union>S\<in>M. X S"]) force
 qed
 
 
 subsubsection \<open>Cauchy filter\<close>
 
 definition cauchy_filter :: "'a filter \<Rightarrow> bool"
   where "cauchy_filter F \<longleftrightarrow> F \<times>\<^sub>F F \<le> uniformity"
 
 definition Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"
   where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)"
 
 lemma Cauchy_uniform_iff:
   "Cauchy X \<longleftrightarrow> (\<forall>P. eventually P uniformity \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)))"
   unfolding Cauchy_uniform cauchy_filter_def le_filter_def eventually_prod_same
     eventually_filtermap eventually_sequentially
 proof safe
   let ?U = "\<lambda>P. eventually P uniformity"
   {
     fix P
     assume "?U P" "\<forall>P. ?U P \<longrightarrow> (\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
     then obtain Q N where "\<And>n. n \<ge> N \<Longrightarrow> Q (X n)" "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> P (x, y)"
       by metis
     then show "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m)"
       by blast
   next
     fix P
     assume "?U P" and P: "\<forall>P. ?U P \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. P (X n, X m))"
     then obtain Q where "?U Q" and Q: "\<And>x y z. Q (x, y) \<Longrightarrow> Q (y, z) \<Longrightarrow> P (x, z)"
       by (auto elim: uniformity_transE)
     then have "?U (\<lambda>x. Q x \<and> (\<lambda>(x, y). Q (y, x)) x)"
       unfolding eventually_conj_iff by (simp add: uniformity_sym)
     from P[rule_format, OF this]
     obtain N where N: "\<And>n m. n \<ge> N \<Longrightarrow> m \<ge> N \<Longrightarrow> Q (X n, X m) \<and> Q (X m, X n)"
       by auto
     show "\<exists>Q. (\<exists>N. \<forall>n\<ge>N. Q (X n)) \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
     proof (safe intro!: exI[of _ "\<lambda>x. \<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)"] exI[of _ N] N)
       fix x y
       assume "\<forall>n\<ge>N. Q (x, X n) \<and> Q (X n, x)" "\<forall>n\<ge>N. Q (y, X n) \<and> Q (X n, y)"
       then have "Q (x, X N)" "Q (X N, y)" by auto
       then show "P (x, y)"
         by (rule Q)
     qed
   }
 qed
 
 lemma nhds_imp_cauchy_filter:
   assumes *: "F \<le> nhds x"
   shows "cauchy_filter F"
 proof -
   have "F \<times>\<^sub>F F \<le> nhds x \<times>\<^sub>F nhds x"
     by (intro prod_filter_mono *)
   also have "\<dots> \<le> uniformity"
     unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same
   proof safe
     fix P
     assume "eventually P uniformity"
     then obtain Ql where ev: "eventually Ql uniformity"
       and "Ql (x, y) \<Longrightarrow> Ql (y, z) \<Longrightarrow> P (x, z)" for x y z
       by (rule uniformity_transE) simp
     with ev[THEN uniformity_sym]
     show "\<exists>Q. eventually (\<lambda>(x', y). x' = x \<longrightarrow> Q y) uniformity \<and>
         (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y))"
       by (rule_tac exI[of _ "\<lambda>y. Ql (y, x) \<and> Ql (x, y)"]) (fastforce elim: eventually_elim2)
   qed
   finally show ?thesis
     by (simp add: cauchy_filter_def)
 qed
 
 lemma LIMSEQ_imp_Cauchy: "X \<longlonglongrightarrow> x \<Longrightarrow> Cauchy X"
   unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter)
 
 lemma Cauchy_subseq_Cauchy:
   assumes "Cauchy X" "strict_mono f"
   shows "Cauchy (X \<circ> f)"
   unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def
   by (rule order_trans[OF _ \<open>Cauchy X\<close>[unfolded Cauchy_uniform cauchy_filter_def]])
      (intro prod_filter_mono filtermap_mono filterlim_subseq[OF \<open>strict_mono f\<close>, unfolded filterlim_def])
 
 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
   unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy)
 
 definition complete :: "'a set \<Rightarrow> bool"
   where complete_uniform: "complete S \<longleftrightarrow>
     (\<forall>F \<le> principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x))"
 
 end
 
 
 subsubsection \<open>Uniformly continuous functions\<close>
 
 definition uniformly_continuous_on :: "'a set \<Rightarrow> ('a::uniform_space \<Rightarrow> 'b::uniform_space) \<Rightarrow> bool"
   where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f \<longleftrightarrow>
     (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)"
 
 lemma uniformly_continuous_onD:
   "uniformly_continuous_on s f \<Longrightarrow> eventually E uniformity \<Longrightarrow>
     eventually (\<lambda>(x, y). x \<in> s \<longrightarrow> y \<in> s \<longrightarrow> E (f x, f y)) uniformity"
   by (simp add: uniformly_continuous_on_uniformity filterlim_iff
       eventually_inf_principal split_beta' mem_Times_iff imp_conjL)
 
 lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. c)"
   by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl)
 
 lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (\<lambda>x. x)"
   by (auto simp: uniformly_continuous_on_uniformity filterlim_def)
 
 lemma uniformly_continuous_on_compose[continuous_intros]:
   "uniformly_continuous_on s g \<Longrightarrow> uniformly_continuous_on (g`s) f \<Longrightarrow>
     uniformly_continuous_on s (\<lambda>x. f (g x))"
   using filterlim_compose[of "\<lambda>(x, y). (f x, f y)" uniformity
       "uniformity_on (g`s)"  "\<lambda>(x, y). (g x, g y)" "uniformity_on s"]
   by (simp add: split_beta' uniformly_continuous_on_uniformity
       filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff)
 
 lemma uniformly_continuous_imp_continuous:
   assumes f: "uniformly_continuous_on s f"
   shows "continuous_on s f"
   by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def
            elim: eventually_mono dest!: uniformly_continuous_onD[OF f])
 
 
 section \<open>Product Topology\<close>
 
 subsection \<open>Product is a topological space\<close>
 
 instantiation prod :: (topological_space, topological_space) topological_space
 begin
 
 definition open_prod_def[code del]:
   "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
     (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
 
 lemma open_prod_elim:
   assumes "open S" and "x \<in> S"
   obtains A B where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
   using assms unfolding open_prod_def by fast
 
 lemma open_prod_intro:
   assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S"
   shows "open S"
   using assms unfolding open_prod_def by fast
 
 instance
 proof
   show "open (UNIV :: ('a \<times> 'b) set)"
     unfolding open_prod_def by auto
 next
   fix S T :: "('a \<times> 'b) set"
   assume "open S" "open T"
   show "open (S \<inter> T)"
   proof (rule open_prod_intro)
     fix x
     assume x: "x \<in> S \<inter> T"
     from x have "x \<in> S" by simp
     obtain Sa Sb where A: "open Sa" "open Sb" "x \<in> Sa \<times> Sb" "Sa \<times> Sb \<subseteq> S"
       using \<open>open S\<close> and \<open>x \<in> S\<close> by (rule open_prod_elim)
     from x have "x \<in> T" by simp
     obtain Ta Tb where B: "open Ta" "open Tb" "x \<in> Ta \<times> Tb" "Ta \<times> Tb \<subseteq> T"
       using \<open>open T\<close> and \<open>x \<in> T\<close> by (rule open_prod_elim)
     let ?A = "Sa \<inter> Ta" and ?B = "Sb \<inter> Tb"
     have "open ?A \<and> open ?B \<and> x \<in> ?A \<times> ?B \<and> ?A \<times> ?B \<subseteq> S \<inter> T"
       using A B by (auto simp add: open_Int)
     then show "\<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S \<inter> T"
       by fast
   qed
 next
   fix K :: "('a \<times> 'b) set set"
   assume "\<forall>S\<in>K. open S"
   then show "open (\<Union>K)"
     unfolding open_prod_def by fast
 qed
 
 end
 
 declare [[code abort: "open :: ('a::topological_space \<times> 'b::topological_space) set \<Rightarrow> bool"]]
 
 lemma open_Times: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<times> T)"
   unfolding open_prod_def by auto
 
 lemma fst_vimage_eq_Times: "fst -` S = S \<times> UNIV"
   by auto
 
 lemma snd_vimage_eq_Times: "snd -` S = UNIV \<times> S"
   by auto
 
 lemma open_vimage_fst: "open S \<Longrightarrow> open (fst -` S)"
   by (simp add: fst_vimage_eq_Times open_Times)
 
 lemma open_vimage_snd: "open S \<Longrightarrow> open (snd -` S)"
   by (simp add: snd_vimage_eq_Times open_Times)
 
 lemma closed_vimage_fst: "closed S \<Longrightarrow> closed (fst -` S)"
   unfolding closed_open vimage_Compl [symmetric]
   by (rule open_vimage_fst)
 
 lemma closed_vimage_snd: "closed S \<Longrightarrow> closed (snd -` S)"
   unfolding closed_open vimage_Compl [symmetric]
   by (rule open_vimage_snd)
 
 lemma closed_Times: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
 proof -
   have "S \<times> T = (fst -` S) \<inter> (snd -` T)"
     by auto
   then show "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<times> T)"
     by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
 qed
 
 lemma subset_fst_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> y \<in> B \<Longrightarrow> A \<subseteq> fst ` S"
   unfolding image_def subset_eq by force
 
 lemma subset_snd_imageI: "A \<times> B \<subseteq> S \<Longrightarrow> x \<in> A \<Longrightarrow> B \<subseteq> snd ` S"
   unfolding image_def subset_eq by force
 
 lemma open_image_fst:
   assumes "open S"
   shows "open (fst ` S)"
 proof (rule openI)
   fix x
   assume "x \<in> fst ` S"
   then obtain y where "(x, y) \<in> S"
     by auto
   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
     using \<open>open S\<close> unfolding open_prod_def by auto
   from \<open>A \<times> B \<subseteq> S\<close> \<open>y \<in> B\<close> have "A \<subseteq> fst ` S"
     by (rule subset_fst_imageI)
   with \<open>open A\<close> \<open>x \<in> A\<close> have "open A \<and> x \<in> A \<and> A \<subseteq> fst ` S"
     by simp
   then show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> fst ` S" ..
 qed
 
 lemma open_image_snd:
   assumes "open S"
   shows "open (snd ` S)"
 proof (rule openI)
   fix y
   assume "y \<in> snd ` S"
   then obtain x where "(x, y) \<in> S"
     by auto
   then obtain A B where "open A" "open B" "x \<in> A" "y \<in> B" "A \<times> B \<subseteq> S"
     using \<open>open S\<close> unfolding open_prod_def by auto
   from \<open>A \<times> B \<subseteq> S\<close> \<open>x \<in> A\<close> have "B \<subseteq> snd ` S"
     by (rule subset_snd_imageI)
   with \<open>open B\<close> \<open>y \<in> B\<close> have "open B \<and> y \<in> B \<and> B \<subseteq> snd ` S"
     by simp
   then show "\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> snd ` S" ..
 qed
 
 lemma nhds_prod: "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
   unfolding nhds_def
 proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal)
   fix S T
   assume "open S" "a \<in> S" "open T" "b \<in> T"
   then show "(INF x \<in> {S. open S \<and> (a, b) \<in> S}. principal x) \<le> principal (S \<times> T)"
     by (intro INF_lower) (auto intro!: open_Times)
 next
   fix S'
   assume "open S'" "(a, b) \<in> S'"
   then obtain S T where "open S" "a \<in> S" "open T" "b \<in> T" "S \<times> T \<subseteq> S'"
     by (auto elim: open_prod_elim)
   then show "(INF x \<in> {S. open S \<and> a \<in> S}. INF y \<in> {S. open S \<and> b \<in> S}.
       principal (x \<times> y)) \<le> principal S'"
     by (auto intro!: INF_lower2)
 qed
 
 
 subsubsection \<open>Continuity of operations\<close>
 
 lemma tendsto_fst [tendsto_intros]:
   assumes "(f \<longlongrightarrow> a) F"
   shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
 proof (rule topological_tendstoI)
   fix S
   assume "open S" and "fst a \<in> S"
   then have "open (fst -` S)" and "a \<in> fst -` S"
     by (simp_all add: open_vimage_fst)
   with assms have "eventually (\<lambda>x. f x \<in> fst -` S) F"
     by (rule topological_tendstoD)
   then show "eventually (\<lambda>x. fst (f x) \<in> S) F"
     by simp
 qed
 
 lemma tendsto_snd [tendsto_intros]:
   assumes "(f \<longlongrightarrow> a) F"
   shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
 proof (rule topological_tendstoI)
   fix S
   assume "open S" and "snd a \<in> S"
   then have "open (snd -` S)" and "a \<in> snd -` S"
     by (simp_all add: open_vimage_snd)
   with assms have "eventually (\<lambda>x. f x \<in> snd -` S) F"
     by (rule topological_tendstoD)
   then show "eventually (\<lambda>x. snd (f x) \<in> S) F"
     by simp
 qed
 
 lemma tendsto_Pair [tendsto_intros]:
   assumes "(f \<longlongrightarrow> a) F" and "(g \<longlongrightarrow> b) F"
   shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
   unfolding nhds_prod using assms by (rule filterlim_Pair)
 
 lemma continuous_fst[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. fst (f x))"
   unfolding continuous_def by (rule tendsto_fst)
 
 lemma continuous_snd[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. snd (f x))"
   unfolding continuous_def by (rule tendsto_snd)
 
 lemma continuous_Pair[continuous_intros]:
   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. (f x, g x))"
   unfolding continuous_def by (rule tendsto_Pair)
 
 lemma continuous_on_fst[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. fst (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_fst)
 
 lemma continuous_on_snd[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. snd (f x))"
   unfolding continuous_on_def by (auto intro: tendsto_snd)
 
 lemma continuous_on_Pair[continuous_intros]:
   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. (f x, g x))"
   unfolding continuous_on_def by (auto intro: tendsto_Pair)
 
 lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap"
   by (simp add: prod.swap_def continuous_on_fst continuous_on_snd
       continuous_on_Pair continuous_on_id)
 
 lemma continuous_on_swap_args:
   assumes "continuous_on (A\<times>B) (\<lambda>(x,y). d x y)"
     shows "continuous_on (B\<times>A) (\<lambda>(x,y). d y x)"
 proof -
   have "(\<lambda>(x,y). d y x) = (\<lambda>(x,y). d x y) \<circ> prod.swap"
     by force
   then show ?thesis
     by (metis assms continuous_on_compose continuous_on_swap product_swap)
 qed
 
 lemma isCont_fst [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. fst (f x)) a"
   by (fact continuous_fst)
 
 lemma isCont_snd [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. snd (f x)) a"
   by (fact continuous_snd)
 
 lemma isCont_Pair [simp]: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) a"
   by (fact continuous_Pair)
 
 lemma continuous_on_compose_Pair:
   assumes f: "continuous_on (Sigma A B) (\<lambda>(a, b). f a b)"
   assumes g: "continuous_on C g"
   assumes h: "continuous_on C h"
   assumes subset: "\<And>c. c \<in> C \<Longrightarrow> g c \<in> A" "\<And>c. c \<in> C \<Longrightarrow> h c \<in> B (g c)"
   shows "continuous_on C (\<lambda>c. f (g c) (h c))"
   using continuous_on_compose2[OF f continuous_on_Pair[OF g h]] subset
   by auto
 
 
 subsubsection \<open>Connectedness of products\<close>
 
 proposition connected_Times:
   assumes S: "connected S" and T: "connected T"
   shows "connected (S \<times> T)"
 proof (rule connectedI_const)
   fix P::"'a \<times> 'b \<Rightarrow> bool"
   assume P[THEN continuous_on_compose2, continuous_intros]: "continuous_on (S \<times> T) P"
   have "continuous_on S (\<lambda>s. P (s, t))" if "t \<in> T" for t
     by (auto intro!: continuous_intros that)
   from connectedD_const[OF S this]
   obtain c1 where c1: "\<And>s t. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> P (s, t) = c1 t"
     by metis
   moreover
   have "continuous_on T (\<lambda>t. P (s, t))" if "s \<in> S" for s
     by (auto intro!: continuous_intros that)
   from connectedD_const[OF T this]
   obtain c2 where "\<And>s t. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> P (s, t) = c2 s"
     by metis
   ultimately show "\<exists>c. \<forall>s\<in>S \<times> T. P s = c"
     by auto
 qed
 
 corollary connected_Times_eq [simp]:
    "connected (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> connected S \<and> connected T"  (is "?lhs = ?rhs")
 proof
   assume L: ?lhs
   show ?rhs
   proof cases
     assume "S \<noteq> {} \<and> T \<noteq> {}"
     moreover
     have "connected (fst ` (S \<times> T))" "connected (snd ` (S \<times> T))"
       using continuous_on_fst continuous_on_snd continuous_on_id
       by (blast intro: connected_continuous_image [OF _ L])+
     ultimately show ?thesis
       by auto
   qed auto
 qed (auto simp: connected_Times)
 
 
 subsubsection \<open>Separation axioms\<close>
 
 instance prod :: (t0_space, t0_space) t0_space
 proof
   fix x y :: "'a \<times> 'b"
   assume "x \<noteq> y"
   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
   then show "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
     by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
 qed
 
 instance prod :: (t1_space, t1_space) t1_space
 proof
   fix x y :: "'a \<times> 'b"
   assume "x \<noteq> y"
   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
   then show "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
     by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
 qed
 
 instance prod :: (t2_space, t2_space) t2_space
 proof
   fix x y :: "'a \<times> 'b"
   assume "x \<noteq> y"
   then have "fst x \<noteq> fst y \<or> snd x \<noteq> snd y"
     by (simp add: prod_eq_iff)
   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
     by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
 qed
 
 lemma isCont_swap[continuous_intros]: "isCont prod.swap a"
   using continuous_on_eq_continuous_within continuous_on_swap by blast
 
 lemma open_diagonal_complement:
   "open {(x,y) |x y. x \<noteq> (y::('a::t2_space))}"
 proof -
   have "open {(x, y). x \<noteq> (y::'a)}"
     unfolding split_def by (intro open_Collect_neq continuous_intros)
   also have "{(x, y). x \<noteq> (y::'a)} = {(x, y) |x y. x \<noteq> (y::'a)}"
     by auto
   finally show ?thesis .
 qed
 
 lemma closed_diagonal:
   "closed {y. \<exists> x::('a::t2_space). y = (x,x)}"
 proof -
   have "{y. \<exists> x::'a. y = (x,x)} = UNIV - {(x,y) | x y. x \<noteq> y}" by auto
   then show ?thesis using open_diagonal_complement closed_Diff by auto
 qed
 
 lemma open_superdiagonal:
   "open {(x,y) | x y. x > (y::'a::{linorder_topology})}"
 proof -
   have "open {(x, y). x > (y::'a)}"
     unfolding split_def by (intro open_Collect_less continuous_intros)
   also have "{(x, y). x > (y::'a)} = {(x, y) |x y. x > (y::'a)}"
     by auto
   finally show ?thesis .
 qed
 
 lemma closed_subdiagonal:
   "closed {(x,y) | x y. x \<le> (y::'a::{linorder_topology})}"
 proof -
   have "{(x,y) | x y. x \<le> (y::'a)} = UNIV - {(x,y) | x y. x > (y::'a)}" by auto
   then show ?thesis using open_superdiagonal closed_Diff by auto
 qed
 
 lemma open_subdiagonal:
   "open {(x,y) | x y. x < (y::'a::{linorder_topology})}"
 proof -
   have "open {(x, y). x < (y::'a)}"
     unfolding split_def by (intro open_Collect_less continuous_intros)
   also have "{(x, y). x < (y::'a)} = {(x, y) |x y. x < (y::'a)}"
     by auto
   finally show ?thesis .
 qed
 
 lemma closed_superdiagonal:
   "closed {(x,y) | x y. x \<ge> (y::('a::{linorder_topology}))}"
 proof -
   have "{(x,y) | x y. x \<ge> (y::'a)} = UNIV - {(x,y) | x y. x < y}" by auto
   then show ?thesis using open_subdiagonal closed_Diff by auto
 qed
 
 end