src/HOL/Analysis/Elementary_Topology.thy

tuned defs

(*  Author:     L C Paulson, University of Cambridge
    Author:     Amine Chaieb, University of Cambridge
    Author:     Robert Himmelmann, TU Muenchen
    Author:     Brian Huffman, Portland State University
*)

section \<open>Elementary Topology\<close>

theory Elementary_Topology
imports
  "HOL-Library.Set_Idioms"
  Product_Vector
begin


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


subsection \<open>Topological Basis\<close>

context topological_space
begin

definition%important "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%important countable_basis =
  fixes B :: "'a::topological_space set set"
  assumes is_basis: "topological_basis B"
    and countable_basis: "countable B"
begin

lemma open_countable_basis_ex:
  assumes "open 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 "open 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. open 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. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
  using countable_dense_exists by blast

end

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::topological_space 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_space_class.topological_basis_def
      by (safe intro!: topological_space_class.open_Inter)
         (simp_all add: B generate_topology.Basis subset_eq)
  qed
qed

end

sublocale second_countable_topology <
  countable_basis "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%important \<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%important (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%important 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


subsection \<open>Closure of a Set\<close>

definition%important 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%important 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%unimportant \<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)


subsection \<open>Limits\<close>

lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
  by (rule topological_tendstoI) (auto elim: eventually_mono)

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>Deducing things about the limit from the elements.\<close>

lemma Lim_in_closed_set:
  assumes "closed S"
    and "eventually (\<lambda>x. f(x) \<in> S) net"
    and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
  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) net"
    by (rule topological_tendstoD)
  with assms(2) have "eventually (\<lambda>x. False) net"
    by (rule eventually_elim2) simp
  with assms(3) show "False"
    by (simp add: eventually_False)
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_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
  by (rule tendsto_Lim) (auto intro: tendsto_intros)

lemma netlimit_at [simp]:
  fixes a :: "'a::{perfect_space,t2_space}"
  shows "netlimit (at a) = a"
  using netlimit_within [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>

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: set_mp[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%important 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%important 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 \<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

end