Theory HOL-Analysis.Elementary_Topology

(*  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 ‹Topology›

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

section ‹Elementary Topology›


subsubsection✐‹tag unimportant› ‹Affine transformations of intervals›

lemma real_affinity_le: "0 < m ⟹ m * x + c ≤ y ⟷ x ≤ inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_le_affinity: "0 < m ⟹ y ≤ m * x + c ⟷ inverse m * y + - (c / m) ≤ x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_affinity_lt: "0 < m ⟹ m * x + c < y ⟷ x < inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_lt_affinity: "0 < m ⟹ y < m * x + c ⟷ inverse m * y + - (c / m) < x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_affinity_eq: "m ≠ 0 ⟹ m * x + c = y ⟷ x = inverse m * y + - (c / m)"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)

lemma real_eq_affinity: "m ≠ 0 ⟹ y = m * x + c  ⟷ inverse m * y + - (c / m) = x"
  for m :: "'a::linordered_field"
  by (simp add: field_simps)


subsection ‹Topological Basis›

context topological_space
begin

definition✐‹tag important› "topological_basis B ⟷
  (∀b∈B. open b) ∧ (∀x. open x ⟶ (∃B'. B' ⊆ B ∧ ⋃B' = x))"

lemma topological_basis:
  "topological_basis B ⟷ (∀x. open x ⟷ (∃B'. B' ⊆ B ∧ ⋃B' = x))"
    (is "?lhs = ?rhs")
proof
  show "?lhs ⟹ ?rhs"
    by (meson local.open_Union subsetD topological_basis_def)
  show "?rhs ⟹ ?lhs"
    unfolding topological_basis_def
    by (metis cSup_singleton empty_subsetI insert_subset)
qed

lemma topological_basis_iff:
  assumes "⋀B'. B' ∈ B ⟹ open B'"
  shows "topological_basis B ⟷ (∀O'. open O' ⟶ (∀x∈O'. ∃B'∈B. x ∈ B' ∧ B' ⊆ O'))"
    (is "_ ⟷ ?rhs")
proof safe
  fix O' and x::'a
  assume H: "topological_basis B" "open O'" "x ∈ O'"
  then have "(∃B'⊆B. ⋃B' = O')" by (simp add: topological_basis_def)
  then obtain B' where "B' ⊆ B" "O' = ⋃B'" by auto
  then show "∃B'∈B. x ∈ B' ∧ B' ⊆ 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 "∀x∈O'. f x ∈ B ∧ x ∈ f x ∧ f x ⊆ O'"
      by (force intro: bchoice simp: Bex_def)
    then show "∃B'⊆B. ⋃B' = O'"
      by (auto intro: exI[where x="{f x |x. x ∈ O'}"])
  qed
qed

lemma topological_basisI:
  assumes "⋀B'. B' ∈ B ⟹ open B'"
    and "⋀O' x. open O' ⟹ x ∈ O' ⟹ ∃B'∈B. x ∈ B' ∧ B' ⊆ O'"
  shows "topological_basis B"
  by (simp add: assms topological_basis_iff)

lemma topological_basisE:
  fixes O'
  assumes "topological_basis B"
    and "open O'"
    and "x ∈ O'"
  obtains B' where "B' ∈ B" "x ∈ B'" "B' ⊆ O'"
  by (metis assms topological_basis_def topological_basis_iff)

lemma topological_basis_open:
  assumes "topological_basis B" and "X ∈ 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' ⊆ B" "S = ⋃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 ⇒ 'a"
  assumes "topological_basis B" and "⋀B'. B' ≠ {} ⟹ f B' ∈ B'"
  shows "∀X. open X ⟶ X ≠ {} ⟶ (∃B' ∈ B. f B' ∈ X)"
  by (metis assms equals0D in_mono topological_basisE)

end

lemma topological_basis_prod:
  assumes A: "topological_basis A"
    and B: "topological_basis B"
  shows "topological_basis ((λ(a, b). a × b) ` (A × B))"
proof -
  have "∃X⊆A × B. (⋃(a,b)∈X. a × b) = S" if "open S" for S
  proof -
    have "(x, y) ∈ (⋃(a, b)∈{X ∈ A × B. fst X × snd X ⊆ S}. a × b)"
      if xy: "(x, y) ∈ S" for x y
    proof -
      obtain a b where a: "open a""x ∈ a" and b: "open b" "y ∈ b" and "a × b ⊆ S"
        by (metis open_prod_elim[OF ‹open S›] xy mem_Sigma_iff)
      moreover obtain A0 where "A0 ∈ A" "x ∈ A0" "A0 ⊆ a"
        using A a b topological_basisE by blast
      moreover
      from B b obtain B0 where "B0 ∈ B" "y ∈ B0" "B0 ⊆ b"
        by (rule topological_basisE)
      ultimately show ?thesis
        by (intro UN_I[of "(A0, B0)"]) auto
    qed
    then have "(⋃(a, b)∈{x ∈ A × B. fst x × snd x ⊆ S}. a × b) = S"
      by force
    then show ?thesis
      using subset_eq by force
  qed
  with A B show ?thesis
    unfolding topological_basis_def
    by (smt (verit) SigmaE imageE image_mono open_Times case_prod_conv)
qed


subsection ‹Countable Basis›

locale✐‹tag important› countable_basis = topological_space p for p::"'a set ⇒ 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 "∃B' ⊆ B. X = ⋃B'"
  using assms countable_basis is_basis
  unfolding topological_basis_def by blast

lemma open_countable_basisE:
  assumes "p X"
  obtains B' where "B' ⊆ B" "X = ⋃B'"
  using assms open_countable_basis_ex by auto

lemma countable_dense_exists:
  "∃D::'a set. countable D ∧ (∀X. p X ⟶ X ≠ {} ⟶ (∃d ∈ D. d ∈ X))"
proof -
  let ?f = "(λB'. SOME x. x ∈ 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" "⋀X. p X ⟹ X ≠ {} ⟹ ∃d ∈ D. d ∈ 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" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
    "⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)"
proof -
  obtain 𝒜 where 𝒜: "(∀i::nat. x ∈ 𝒜 i ∧ open (𝒜 i))" "(∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
    using first_countable_basis[of x] by metis
  moreover have "countable (range 𝒜)"
    by simp
  ultimately show thesis
    by (smt (verit, best) imageE rangeI that)
qed

lemma (in first_countable_topology) first_countable_basis_Int_stableE:
  obtains 𝒜 where "countable 𝒜" "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
    "⋀S. open S ⟹ x ∈ S ⟹ (∃A∈𝒜. A ⊆ S)"
    "⋀A B. A ∈ 𝒜 ⟹ B ∈ 𝒜 ⟹ A ∩ B ∈ 𝒜"
proof atomize_elim
  obtain ℬ where ℬ:
    "countable ℬ"
    "⋀B. B ∈ ℬ ⟹ x ∈ B"
    "⋀B. B ∈ ℬ ⟹ open B"
    "⋀S. open S ⟹ x ∈ S ⟹ ∃B∈ℬ. B ⊆ S"
    by (rule first_countable_basisE) blast
  define 𝒜 where [abs_def]:
    "𝒜 = (λN. ⋂((λn. from_nat_into ℬ n) ` N)) ` (Collect finite::nat set set)"
  then show "∃𝒜. countable 𝒜 ∧ (∀A. A ∈ 𝒜 ⟶ x ∈ A) ∧ (∀A. A ∈ 𝒜 ⟶ open A) ∧
        (∀S. open S ⟶ x ∈ S ⟶ (∃A∈𝒜. A ⊆ S)) ∧ (∀A B. A ∈ 𝒜 ⟶ B ∈ 𝒜 ⟶ A ∩ B ∈ 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 ∈ 𝒜"
    then show "x ∈ A" "open A"
      using ℬ(4)[OF open_UNIV] by (auto simp: 𝒜_def intro: ℬ from_nat_into)
  next
    let ?int = "λN. ⋂(from_nat_into ℬ ` N)"
    fix A B
    assume "A ∈ 𝒜" "B ∈ 𝒜"
    then obtain N M where "A = ?int N" "B = ?int M" "finite (N ∪ M)"
      by (auto simp: 𝒜_def)
    then show "A ∩ B ∈ 𝒜"
      by (auto simp: 𝒜_def intro!: image_eqI[where x="N ∪ M"])
  next
    fix S
    assume "open S" "x ∈ S"
    then obtain a where a: "a∈ℬ" "a ⊆ S" using ℬ by blast
    moreover have "a∈𝒜"
      unfolding 𝒜_def 
    proof (rule image_eqI)
      show "a = ⋂ (from_nat_into ℬ ` {to_nat_on ℬ a})"
        by (simp add: ℬ a)
    qed auto
    ultimately show "∃a∈𝒜. a ⊆ S"
      by blast 
  qed
qed

lemma (in topological_space) first_countableI:
  assumes "countable 𝒜"
    and 1: "⋀A. A ∈ 𝒜 ⟹ x ∈ A" "⋀A. A ∈ 𝒜 ⟹ open A"
    and 2: "⋀S. open S ⟹ x ∈ S ⟹ ∃A∈𝒜. A ⊆ S"
  shows "∃𝒜::nat ⇒ 'a set. (∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
proof (safe intro!: exI[of _ "from_nat_into 𝒜"])
  fix i
  have "𝒜 ≠ {}" using 2[of UNIV] by auto
  show "x ∈ from_nat_into 𝒜 i" "open (from_nat_into 𝒜 i)"
    using range_from_nat_into_subset[OF ‹𝒜 ≠ {}›] 1 by auto
next
  fix S
  assume "open S" "x∈S" 
  then show "∃i. from_nat_into 𝒜 i ⊆ S"
    by (metis "2" ‹countable 𝒜› from_nat_into_surj)
qed

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
proof
  fix x :: "'a × 'b"
  obtain 𝒜 where 𝒜:
      "countable 𝒜"
      "⋀a. a ∈ 𝒜 ⟹ fst x ∈ a"
      "⋀a. a ∈ 𝒜 ⟹ open a"
      "⋀S. open S ⟹ fst x ∈ S ⟹ ∃a∈𝒜. a ⊆ S"
    by (rule first_countable_basisE[of "fst x"]) blast
  obtain B where B:
      "countable B"
      "⋀a. a ∈ B ⟹ snd x ∈ a"
      "⋀a. a ∈ B ⟹ open a"
      "⋀S. open S ⟹ snd x ∈ S ⟹ ∃a∈B. a ⊆ S"
    by (rule first_countable_basisE[of "snd x"]) blast
  show "∃𝒜::nat ⇒ ('a × 'b) set.
    (∀i. x ∈ 𝒜 i ∧ open (𝒜 i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. 𝒜 i ⊆ S))"
  proof (rule first_countableI[of "(λ(a, b). a × b) ` (𝒜 × B)"], safe)
    fix a b
    assume x: "a ∈ 𝒜" "b ∈ B"
    show "x ∈ a × b" 
      by (simp add: 𝒜(2) B(2) mem_Times_iff x)
    show "open (a × b)"
      by (simp add: 𝒜(3) B(3) open_Times x)
  next
    fix S
    assume "open S" "x ∈ S"
    then obtain a' b' where a'b': "open a'" "open b'" "x ∈ a' × b'" "a' × b' ⊆ S"
      by (rule open_prod_elim)
    moreover
    obtain a b where "a ∈ 𝒜" "a ⊆ a'" "b ∈ B" "b ⊆ b'"
      by (meson B(4) 𝒜(4) a'b' mem_Times_iff)
    ultimately
    show "∃a∈(λ(a, b). a × b) ` (𝒜 × B). a ⊆ S"
      by (auto intro!: bexI[of _ "a × b"] bexI[of _ a] bexI[of _ b])
  qed (simp add: 𝒜 B)
qed

class second_countable_topology = topological_space +
  assumes ex_countable_subbasis:
    "∃B::'a set set. countable B ∧ open = generate_topology B"
begin

lemma ex_countable_basis: "∃B::'a set set. countable B ∧ 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 ∧ b ⊆ 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 "∃B'⊆{b. finite b ∧ b ⊆ B}. (⋃b∈B'. ⋂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 = ⋃(Inter ` x)" "⋀i. i ∈ x ⟹ finite i ∧ i ⊆ B"
          and y: "b = ⋃(Inter ` y)" "⋀i. i ∈ y ⟹ finite i ∧ i ⊆ B"
          by blast
        show ?case
          unfolding x y Int_UN_distrib2
          by (intro exI[of _ "{i ∪ j| i j.  i ∈ x ∧ j ∈ y}"]) (auto dest: x(2) y(2))
      next
        case (UN K)
        then have "∀k∈K. ∃B'⊆{b. finite b ∧ b ⊆ B}. ⋃ (Inter ` B') = k" by auto
        then obtain k where
            "∀ka∈K. k ka ⊆ {b. finite b ∧ b ⊆ B} ∧ ⋃(Inter ` (k ka)) = ka"
          unfolding bchoice_iff ..
        then show "∃B'⊆{b. finite b ∧ b ⊆ B}. ⋃ (Inter ` B') = ⋃K"
          by (intro exI[of _ "⋃(k ` K)"]) auto
      next
        case (Basis S)
        then show ?case
          by (intro exI[of _ "{{S}}"]) auto
      qed
      then have "(∃B'⊆Inter ` {b. finite b ∧ b ⊆ B}. ⋃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" "⋀C. C ∈ ℬ ⟹ open C"
       "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
by (metis ex_countable_basis topological_basis_def)

proposition Lindelof:
  fixes ℱ :: "'a::second_countable_topology set set"
  assumes ℱ: "⋀S. S ∈ ℱ ⟹ open S"
  obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
  obtain ℬ :: "'a set set"
    where "countable ℬ" "⋀C. C ∈ ℬ ⟹ open C"
      and ℬ: "⋀S. open S ⟹ ∃U. U ⊆ ℬ ∧ S = ⋃U"
    using univ_second_countable by blast
  define 𝒟 where "𝒟 ≡ {S. S ∈ ℬ ∧ (∃U. U ∈ ℱ ∧ S ⊆ U)}"
  have "countable 𝒟"
    by (simp add: 𝒟_def ‹countable ℬ›)
  have "⋀S. ∃U. S ∈ 𝒟 ⟶ U ∈ ℱ ∧ S ⊆ U"
    by (simp add: 𝒟_def)
  then obtain G where G: "⋀S. S ∈ 𝒟 ⟶ G S ∈ ℱ ∧ S ⊆ G S"
    by metis
  have "⋃ℱ ⊆ ⋃𝒟"
    unfolding 𝒟_def by (blast dest: ℱ ℬ)
  moreover have "⋃𝒟 ⊆ ⋃ℱ"
    using 𝒟_def by blast
  ultimately have eq1: "⋃ℱ = ⋃𝒟" ..
  moreover have "⋃𝒟 = ⋃ (G ` 𝒟)"
    using G eq1 by auto
  ultimately show ?thesis
    by (metis G ‹countable 𝒟› countable_image image_subset_iff that)
qed

lemma countable_disjoint_open_subsets:
  fixes ℱ :: "'a::second_countable_topology set set"
  assumes "⋀S. S ∈ ℱ ⟹ open S" and pw: "pairwise disjnt ℱ"
    shows "countable ℱ"
proof -
  obtain ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
    by (meson assms Lindelof)
  with pw have "ℱ ⊆ insert {} ℱ'"
    by (fastforce simp add: pairwise_def disjnt_iff)
  then show ?thesis
    by (simp add: ‹countable ℱ'› countable_subset)
qed

sublocale second_countable_topology <
  countable_basis "open" "SOME B. countable B ∧ 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 "∃B::('a × 'b) set set. countable B ∧ open = generate_topology B"
    by (auto intro!: exI[of _ "(λ(a, b). a × b) ` (A × B)"] topological_basis_prod
      topological_basis_imp_subbasis)
qed

instance second_countable_topology ⊆ first_countable_topology
proof
  fix x :: 'a
  define B :: "'a set set" where "B = (SOME B. countable B ∧ 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 "∃A::nat ⇒ 'a set.
    (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S ⟶ (∃i. A i ⊆ S))"
    by (intro first_countableI[of "{b∈B. x ∈ b}"])
       (fastforce simp: topological_space_class.topological_basis_def)+
qed

instance nat :: second_countable_topology
proof
  show "∃B::nat set set. countable B ∧ open = generate_topology B"
    by (intro exI[of _ "range lessThan ∪ range greaterThan"]) (auto simp: open_nat_def)
qed

lemma countable_separating_set_linorder1:
  shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b ≤ 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 ∈ U)| U. U ∈ A}"
  then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image)
  define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}"
  then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image)
  have "∃b ∈ B1 ∪ B2. x < b ∧ b ≤ y" if "x < y" for x y
  proof (cases)
    assume "∃z. x < z ∧ z < y"
    then obtain z where z: "x < z ∧ z < y" by auto
    define U where "U = {x<..<y}"
    then have "open U" by simp
    moreover have "z ∈ U" using z U_def by simp
    ultimately obtain V where "V ∈ A" "z ∈ V" "V ⊆ U" 
      using topological_basisE[OF ‹topological_basis A›] by auto
    define w where "w = (SOME x. x ∈ V)"
    then have "w ∈ V" using ‹z ∈ V› by (metis someI2)
    then have "x < w ∧ w ≤ y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce
    moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto
    ultimately show ?thesis by auto
  next
    assume "¬(∃z. x < z ∧ z < y)"
    then have *: "⋀z. z > x ⟹ z ≥ y" by auto
    define U where "U = {x<..}"
    then have "open U" by simp
    moreover have "y ∈ U" using ‹x < y› U_def by simp
    ultimately obtain "V" where "V ∈ A" "y ∈ V" "V ⊆ U" 
      using topological_basisE[OF ‹topological_basis A›] by auto
    have "U = {y..}" unfolding U_def using * ‹x < y› by auto
    then have "V ⊆ {y..}" using ‹V ⊆ U› by simp
    then have "(LEAST w. w ∈ V) = y" using ‹y ∈ V› by (meson Least_equality atLeast_iff subsetCE)
    then have "y ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto
    moreover have "x < y ∧ y ≤ y" using ‹x < y› by simp
    ultimately show ?thesis by auto
  qed
  moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp
  ultimately show ?thesis by auto
qed

lemma countable_separating_set_linorder2:
  shows "∃B::('a::{linorder_topology, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x ≤ b ∧ 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 ∈ U) | U. U ∈ A}"
  then have "countable B1" using ‹countable A› by (simp add: Setcompr_eq_image)
  define B2 where "B2 = {(SOME x. x ∈ U)| U. U ∈ A}"
  then have "countable B2" using ‹countable A› by (simp add: Setcompr_eq_image)
  have "∃b ∈ B1 ∪ B2. x ≤ b ∧ b < y" if "x < y" for x y
  proof (cases)
    assume "∃z. x < z ∧ z < y"
    then obtain z where z: "x < z ∧ z < y" by auto
    define U where "U = {x<..<y}"
    then have "open U" by simp
    moreover have "z ∈ U" using z U_def by simp
    ultimately obtain "V" where "V ∈ A" "z ∈ V" "V ⊆ U" 
      using topological_basisE[OF ‹topological_basis A›] by auto
    define w where "w = (SOME x. x ∈ V)"
    then have "w ∈ V" using ‹z ∈ V› by (metis someI2)
    then have "x ≤ w ∧ w < y" using ‹w ∈ V› ‹V ⊆ U› U_def by fastforce
    moreover have "w ∈ B1 ∪ B2" using w_def B2_def ‹V ∈ A› by auto
    ultimately show ?thesis by auto
  next
    assume "¬(∃z. x < z ∧ z < y)"
    then have *: "⋀z. z < y ⟹ z ≤ x" using leI by blast
    define U where "U = {..<y}"
    then have "open U" by simp
    moreover have "x ∈ U" using ‹x < y› U_def by simp
    ultimately obtain "V" where "V ∈ A" "x ∈ V" "V ⊆ U" 
      using topological_basisE[OF ‹topological_basis A›] by auto
    have "U = {..x}" unfolding U_def using * ‹x < y› by auto
    then have "V ⊆ {..x}" using ‹V ⊆ U› by simp
    then have "(GREATEST x. x ∈ V) = x" using ‹x ∈ V› by (meson Greatest_equality atMost_iff subsetCE)
    then have "x ∈ B1 ∪ B2" using ‹V ∈ A› B1_def by auto
    moreover have "x ≤ x ∧ x < y" using ‹x < y› by simp
    ultimately show ?thesis by auto
  qed
  moreover have "countable (B1 ∪ B2)" using ‹countable B1› ‹countable B2› by simp
  ultimately show ?thesis by auto
qed

lemma countable_separating_set_dense_linorder:
  shows "∃B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B ∧ (∀x y. x < y ⟶ (∃b ∈ B. x < b ∧ b < y))"
proof -
  obtain B::"'a set" where B: "countable B" "⋀x y. x < y ⟹ (∃b ∈ B. x < b ∧ b ≤ y)"
    using countable_separating_set_linorder1 by auto
  have "∃b ∈ B. x < b ∧ b < y" if "x < y" for x y
  proof -
    obtain z where "x < z" "z < y" using ‹x < y› dense by blast
    then obtain b where "b ∈ B" "x < b ∧ b ≤ z" using B(2) by auto
    then have "x < b ∧ b < y" using ‹z < y› by auto
    then show ?thesis using ‹b ∈ B› by auto
  qed
  then show ?thesis using B(1) by auto
qed


subsection ‹Polish spaces›

text ‹Textbooks define Polish spaces as completely metrizable.
  We assume the topology to be complete for a given metric.›

class polish_space = complete_space + second_countable_topology


subsection ‹Limit Points›

definition✐‹tag important› (in topological_space) islimpt:: "'a ⇒ 'a set ⇒ bool"  (infixr ‹islimpt› 60)
  where "x islimpt S ⟷ (∀T. x∈T ⟶ open T ⟶ (∃y∈S. y∈T ∧ y≠x))"

lemma islimptI:
  assumes "⋀T. x ∈ T ⟹ open T ⟹ ∃y∈S. y ∈ T ∧ y ≠ x"
  shows "x islimpt S"
  using assms unfolding islimpt_def by auto

lemma islimptE:
  assumes "x islimpt S" and "x ∈ T" and "open T"
  obtains y where "y ∈ S" and "y ∈ T" and "y ≠ x"
  using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S ⟷ ¬ eventually (λy. y ∉ S) (at x)"
  unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "x islimpt S ⟹ S ⊆ T ⟹ x islimpt T"
  unfolding islimpt_def by fast

lemma islimpt_UNIV_iff: "x islimpt UNIV ⟷ ¬ 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 ‹A perfect space has no isolated points.›

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 ⟷ (∀x. x islimpt S ⟶ x ∈ S)"
  unfolding closed_def open_subopen [of "-S"]
  by (metis Compl_iff islimptE islimptI open_subopen subsetI)

lemma islimpt_EMPTY[simp]: "¬ x islimpt {}"
  by (auto simp: islimpt_def)

lemma islimpt_Un: "x islimpt (S ∪ T) ⟷ x islimpt S ∨ x islimpt T"
  by (simp add: islimpt_iff_eventually eventually_conj_iff)

lemma islimpt_finite_union_iff:
  assumes "finite A"
  shows   "z islimpt (⋃x∈A. B x) ⟷ (∃x∈A. z islimpt B x)"
  using assms by (induction rule: finite_induct) (simp_all add: islimpt_Un)

lemma islimpt_insert:
  fixes x :: "'a::t1_space"
  shows "x islimpt (insert a S) ⟷ x islimpt S"
proof
  assume "x islimpt (insert a S)"
  then show "x islimpt S"
    by (metis closed_limpt closed_singleton empty_iff insert_iff insert_is_Un islimpt_Un islimpt_def)
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 ⟹ ¬ x islimpt S"
  by (induct set: finite) (simp_all add: islimpt_insert)

lemma islimpt_Un_finite:
  fixes x :: "'a::t1_space"
  shows "finite S ⟹ x islimpt (S ∪ T) ⟷ x islimpt T"
  by (simp add: islimpt_Un islimpt_finite)

lemma islimpt_eq_acc_point:
  fixes l :: "'a :: t1_space"
  shows "l islimpt S ⟷ (∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S))"
proof (safe intro!: islimptI)
  fix U
  assume "l islimpt S" "l ∈ U" "open U" "finite (U ∩ S)"
  then have "l islimpt S" "l ∈ (U - (U ∩ S - {l}))" "open (U - (U ∩ S - {l}))"
    by (auto intro: finite_imp_closed)
  then show False
    by (rule islimptE) auto
next
  fix T
  assume *: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ S)" "l ∈ T" "open T"
  then have "∃x. x ∈ (T ∩ S - {l})"
    by (metis ex_in_conv finite.emptyI infinite_remove)
  then show "∃y∈S. y ∈ T ∧ y ≠ l"
    by auto
qed

lemma acc_point_range_imp_convergent_subsequence:
  fixes l :: "'a :: first_countable_topology"
  assumes l: "∀U. l∈U ⟶ open U ⟶ infinite (U ∩ range f)"
  shows "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
proof -
  from countable_basis_at_decseq[of l]
  obtain A where A:
      "⋀i. open (A i)"
      "⋀i. l ∈ A i"
      "⋀S. open S ⟹ l ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially"
    by blast
  define s where "s n i = (SOME j. i < j ∧ f j ∈ A (Suc n))" for n i
  {
    fix n i
    have "infinite (A (Suc n) ∩ range f - f`{.. i})"
      using l A by auto
    then have "∃x. x ∈ A (Suc n) ∩ range f - f`{.. i}"
      by (metis all_not_in_conv finite.emptyI)
    then have "∃a. i < a ∧ f a ∈ A (Suc n)"
      by (force simp: linorder_not_le)
    then have "i < s n i" "f (s n i) ∈ 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 "(λn. f (r n)) ⇢ l"
  proof (rule topological_tendstoI)
    fix S
    assume "open S" "l ∈ S"
    with A(3) have "eventually (λi. A i ⊆ S) sequentially"
      by auto
    moreover
    {
      fix i
      assume "Suc 0 ≤ i"
      then have "f (r i) ∈ A i"
        by (cases i) (simp_all add: r_def s)
    }
    then have "eventually (λi. f (r i) ∈ A i) sequentially"
      by (auto simp: eventually_sequentially)
    ultimately show "eventually (λi. f (r i) ∈ S) sequentially"
      by eventually_elim auto
  qed
  ultimately show "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ 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 "∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
  using l unfolding islimpt_eq_acc_point
  by (rule acc_point_range_imp_convergent_subsequence)

lemma sequence_unique_limpt:
  fixes f :: "nat ⇒ 'a::t2_space"
  assumes f: "(f ⤏ l) sequentially"
    and "l' islimpt (range f)"
  shows "l' = l"
proof (rule ccontr)
  assume "l' ≠ l"
  obtain s t where "open s" "open t" "l' ∈ s" "l ∈ t" "s ∩ t = {}"
    using hausdorff [OF ‹l' ≠ l›] by auto
  then obtain N where "∀n≥N. f n ∈ t"
    by (meson f lim_explicit)

  have "UNIV = {..<N} ∪ {N..}"
    by auto
  then have "l' islimpt (f ` ({..<N} ∪ {N..}))"
    using assms(2) by simp
  then have "l' islimpt (f ` {..<N} ∪ f ` {N..})"
    by (simp add: image_Un)
  then have "l' islimpt (f ` {N..})"
    by (simp add: islimpt_Un_finite)
  then obtain y where "y ∈ f ` {N..}" "y ∈ s" "y ≠ l'"
    using ‹l' ∈ s› ‹open s› by (rule islimptE)
  then obtain n where "N ≤ n" "f n ∈ s" "f n ≠ l'"
    by auto
  with ‹∀n≥N. f n ∈ t› ‹s ∩ t = {}› show False
    by blast
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 ⟷ (∃f. (∀n::nat. f n ∈ S - {x}) ∧ (f ⤏ x) sequentially)"
    (is "?lhs = ?rhs")
proof
  assume ?lhs
  from countable_basis_at_decseq[of x] obtain A where A:
      "⋀i. open (A i)"
      "⋀i. x ∈ A i"
      "⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially"
    by blast
  define f where "f n = (SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y)" for n
  {
    fix n
    from ‹?lhs› have "∃y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"
      unfolding islimpt_def using A(1,2)[of n] by auto
    then have "f n ∈ S ∧ f n ∈ A n ∧ x ≠ f n"
      unfolding f_def by (rule someI_ex)
    then have "f n ∈ S" "f n ∈ A n" "x ≠ f n" by auto
  }
  then have "∀n. f n ∈ S - {x}" by auto
  moreover have "(λn. f n) ⇢ x"
  proof (rule topological_tendstoI)
    fix S
    assume "open S" "x ∈ S"
    from A(3)[OF this] ‹⋀n. f n ∈ A n›
    show "eventually (λx. f x ∈ S) sequentially"
      by (auto elim!: eventually_mono)
  qed
  ultimately show ?rhs by fast
next
  assume ?rhs
  then obtain f :: "nat ⇒ 'a" where f: "⋀n. f n ∈ S - {x}" and lim: "f ⇢ x"
    by auto
  show ?lhs
    unfolding islimpt_def
  proof safe
    fix T
    assume "open T" "x ∈ T"
    from lim[THEN topological_tendstoD, OF this] f
    show "∃y∈S. y ∈ T ∧ y ≠ x"
      unfolding eventually_sequentially by auto
  qed
qed

lemma islimpt_isCont_image:
  fixes f :: "'a :: {first_countable_topology, t2_space} ⇒ 'b :: {first_countable_topology, t2_space}"
  assumes "x islimpt A" and "isCont f x" and ev: "eventually (λy. f y ≠ f x) (at x)"
  shows   "f x islimpt f ` A"
proof -
  from assms(1) obtain g where g: "g ⇢ x" "range g ⊆ A - {x}"
    unfolding islimpt_sequential by blast
  have "filterlim g (at x) sequentially"
    using g by (auto simp: filterlim_at intro!: always_eventually)
  then obtain N where N: "⋀n. n ≥ N ⟹ f (g n) ≠ f x"
    by (metis (mono_tags, lifting) ev eventually_at_top_linorder filterlim_iff)
  have "(λx. g (x + N)) ⇢ x"
    using g(1) by (rule LIMSEQ_ignore_initial_segment)
  hence "(λx. f (g (x + N))) ⇢ f x"
    using assms(2) isCont_tendsto_compose by blast
  moreover have "range (λx. f (g (x + N))) ⊆ f ` A - {f x}"
    using g(2) N by auto
  ultimately show ?thesis
    unfolding islimpt_sequential by (intro exI[of _ "λx. f (g (x + N))"]) auto
qed

lemma islimpt_image:
  assumes "z islimpt g -` A ∩ B" "g z ∉ A" "z ∈ B" "continuous_on B g"
  shows   "g z islimpt A"
  by (smt (verit, best) IntD1 assms continuous_on_topological inf_le2 islimpt_def subset_eq vimageE)
  

subsection ‹Interior of a Set›

definition✐‹tag important› interior :: "('a::topological_space) set ⇒ 'a set" where
"interior S = ⋃{T. open T ∧ T ⊆ S}"

lemma interiorI [intro?]:
  assumes "open T" and "x ∈ T" and "T ⊆ S"
  shows "x ∈ interior S"
  using assms unfolding interior_def by fast

lemma interiorE [elim?]:
  assumes "x ∈ interior S"
  obtains T where "open T" and "x ∈ T" and "T ⊆ 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 ⊆ S"
  by (auto simp: interior_def)

lemma interior_maximal: "T ⊆ S ⟹ open T ⟹ T ⊆ interior S"
  by (auto simp: interior_def)

lemma interior_open: "open S ⟹ interior S = S"
  by (intro equalityI interior_subset interior_maximal subset_refl)

lemma interior_eq: "interior S = S ⟷ open S"
  by (metis open_interior interior_open)

lemma open_subset_interior: "open S ⟹ S ⊆ interior T ⟷ S ⊆ 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 ⊆ T ⟹ interior S ⊆ interior T"
  by (auto simp: interior_def)

lemma interior_unique:
  assumes "T ⊆ S" and "open T"
  assumes "⋀T'. T' ⊆ S ⟹ open T' ⟹ T' ⊆ 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"
  by (meson interior_eq interior_subset not_open_singleton subset_singletonD)

lemma interior_Int [simp]: "interior (S ∩ T) = interior S ∩ interior T"
  by (meson Int_mono Int_subset_iff antisym_conv interior_maximal interior_subset open_Int open_interior)

lemma eventually_nhds_in_nhd: "x ∈ interior s ⟹ eventually (λy. y ∈ 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 ∈ interior S"
  shows "x islimpt S"
proof -
  obtain T where "x ∈ T" "T ⊆ S" "open T"
    using interior_subset x by blast
  with x islimpt_UNIV [of x]  show ?thesis
    unfolding islimpt_def by (metis (full_types) Int_iff open_Int subsetD)
qed

lemma open_imp_islimpt:
  fixes x::"'a:: perfect_space"
  assumes "open S" "x∈S"
  shows "x islimpt S"
  using assms interior_eq interior_limit_point by auto

lemma islimpt_Int_eventually:
  assumes "x islimpt A" "eventually (λy. y ∈ B) (at x)"
  shows   "x islimpt A ∩ B"
  using assms unfolding islimpt_def eventually_at_filter eventually_nhds
  by (metis Int_iff UNIV_I open_Int)

lemma islimpt_conv_frequently_at:
  "x islimpt A ⟷ frequently (λy. y ∈ A) (at x)"
  by (simp add: frequently_def islimpt_iff_eventually)

lemma frequently_at_imp_islimpt:
  assumes "frequently (λy. y ∈ A) (at x)"
  shows   "x islimpt A"
  by (simp add: assms islimpt_conv_frequently_at)  

lemma interior_closed_Un_empty_interior:
  assumes cS: "closed S"
    and iT: "interior T = {}"
  shows "interior (S ∪ T) = interior S"
proof
  show "interior S ⊆ interior (S ∪ T)"
    by (rule interior_mono) (rule Un_upper1)
  show "interior (S ∪ T) ⊆ interior S"
  proof
    fix x
    assume "x ∈ interior (S ∪ T)"
    then obtain R where "open R" "x ∈ R" "R ⊆ S ∪ T" ..
    show "x ∈ interior S"
    proof (rule ccontr)
      assume "x ∉ interior S"
      with ‹x ∈ R› ‹open R› obtain y where "y ∈ R - S"
        unfolding interior_def by fast
      then show False
        by (metis Diff_subset_conv ‹R ⊆ S ∪ T› ‹open R› cS empty_iff iT interiorI open_Diff)
    qed
  qed
qed

lemma interior_Times: "interior (A × B) = interior A × interior B"
proof (rule interior_unique)
  show "interior A × interior B ⊆ A × B"
    by (intro Sigma_mono interior_subset)
  show "open (interior A × interior B)"
    by (intro open_Times open_interior)
  fix T
  assume "T ⊆ A × B" and "open T"
  then show "T ⊆ interior A × interior B"
  proof safe
    fix x y
    assume "(x, y) ∈ T"
    then obtain C D where "open C" "open D" "C × D ⊆ T" "x ∈ C" "y ∈ D"
      using ‹open T› unfolding open_prod_def by fast
    then have "open C" "open D" "C ⊆ A" "D ⊆ B" "x ∈ C" "y ∈ D"
      using ‹T ⊆ A × B› by auto
    then show "x ∈ interior A" and "y ∈ 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 ⊆ {x ..}" "open T"
  moreover have "x ∉ T"
  proof
    assume "x ∈ T"
    obtain y where "y < x" "{y <.. x} ⊆ T"
      using open_left[OF ‹open T› ‹x ∈ T› ‹b < x›] by auto
    with dense[OF ‹y < x›] obtain z where "z ∈ T" "z < x"
      by (auto simp: subset_eq Ball_def)
    with ‹T ⊆ {x ..}› show False by auto
  qed
  ultimately show "T ⊆ {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 ⊆ {.. x}" "open T"
  moreover have "x ∉ T"
  proof
    assume "x ∈ T"
    obtain y where "x < y" "{x ..< y} ⊆ T"
      using open_right[OF ‹open T› ‹x ∈ T› ‹x < b›] by auto
    with dense[OF ‹x < y›] obtain z where "z ∈ T" "x < z"
      by (auto simp: subset_eq Ball_def less_le)
    with ‹T ⊆ {.. x}› show False by auto
  qed
  ultimately show "T ⊆ {..< 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: "⋀S. ⟦S ∈ ℱ; interior S = {}⟧ ⟹ 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 ‹Closure of a Set›

definition✐‹tag important› closure :: "('a::topological_space) set ⇒ 'a set" where
  "closure S = S ∪ {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 ⊆ 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 ⟷ closed S"
  unfolding closure_hull using closed_Inter by (rule hull_eq)

lemma closure_closed [simp]: "closed S ⟹ 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 ⊆ T ⟹ closure S ⊆ closure T"
  unfolding closure_hull by (rule hull_mono)

lemma closure_minimal: "S ⊆ T ⟹ closed T ⟹ closure S ⊆ T"
  unfolding closure_hull by (rule hull_minimal)

lemma closure_unique:
  assumes "S ⊆ T"
    and "closed T"
    and "⋀T'. S ⊆ T' ⟹ closed T' ⟹ T ⊆ 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 ∪ T) = closure S ∪ closure T"
  by (simp add: closure_interior)

lemma closure_eq_empty [iff]: "closure S = {} ⟷ S = {}"
  using closure_empty closure_subset[of S] by blast

lemma closure_subset_eq: "closure S ⊆ S ⟷ closed S"
  using closure_eq[of S] closure_subset[of S] by simp

lemma open_Int_closure_eq_empty: "open S ⟹ (S ∩ closure T) = {} ⟷ S ∩ 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 ⟹ S ∩ closure T ⊆ closure (S ∩ T)"
proof
  fix x
  assume *: "open S" "x ∈ S ∩ closure T"
  then have "x islimpt (S ∩ T)" if "x islimpt T"
    by (metis IntD1 eventually_at_in_open' inf_commute islimpt_Int_eventually that)
  with * show "x ∈ closure (S ∩ 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 × B) = closure A × closure B"
proof (rule closure_unique)
  show "A × B ⊆ closure A × closure B"
    by (intro Sigma_mono closure_subset)
  show "closed (closure A × closure B)"
    by (intro closed_Times closed_closure)
  fix T
  assume "A × B ⊆ T" and "closed T"
  then show "closure A × closure B ⊆ 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 closure_open_Int_superset:
  assumes "open S" "S ⊆ closure T"
  shows "closure(S ∩ T) = closure S"
  by (metis Int_Un_distrib Un_Int_eq(4) assms closure_Un closure_closure open_Int_closure_subset sup.orderE)

lemma closure_Int: "closure (⋂I) ⊆ ⋂{closure S |S. S ∈ I}"
  by (simp add: INF_greatest Inter_lower Setcompr_eq_image closure_mono)

lemma islimpt_in_closure: "(x islimpt S) = (x∈closure(S-{x}))"
  unfolding closure_def using islimpt_punctured by blast

lemma connected_imp_connected_closure: "connected S ⟹ 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 "⋀x. x ∈ A ⟹ m ≤ x"
    by (auto simp: bdd_below_def)
  then have "A ⊆ {m..}" by auto
  then have "closure A ⊆ {m..}"
    using closed_real_atLeast by (rule closure_minimal)
  then show ?thesis
    by (auto simp: bdd_below_def)
qed


subsection ‹Frontier (also known as boundary)›

definition✐‹tag important› frontier :: "('a::topological_space) set ⇒ '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 ∩ closure (- S)"
  by (auto simp: frontier_def interior_closure)

lemma frontier_Int: "frontier(S ∩ T) = closure(S ∩ T) ∩ (frontier S ∪ frontier T)"
proof -
  have "closure (S ∩ T) ⊆ closure S" "closure (S ∩ T) ⊆ closure T"
    by (simp_all add: closure_mono)
  then show ?thesis
    by (auto simp: frontier_closures)
qed

lemma frontier_Int_subset: "frontier(S ∩ T) ⊆ frontier S ∪ frontier T"
  by (auto simp: frontier_Int)

lemma frontier_Int_closed:
  assumes "closed S" "closed T"
  shows "frontier(S ∩ T) = (frontier S ∩ T) ∪ (S ∩ frontier T)"
  by (smt (verit, best) Diff_Int Int_Diff assms closed_Int closure_eq frontier_def inf_commute interior_Int)

lemma frontier_subset_closed: "closed S ⟹ frontier S ⊆ 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 ⊆ S ⟷ closed S"
  by (metis Diff_subset_conv closure_subset_eq frontier_def interior_subset subset_Un_eq)

lemma frontier_complement [simp]: "frontier (- S) = frontier S"
  by (auto simp: frontier_def closure_complement interior_complement)

lemma frontier_Un_subset: "frontier(S ∪ T) ⊆ frontier S ∪ frontier T"
  by (metis compl_sup frontier_Int_subset frontier_complement)

lemma frontier_disjoint_eq: "frontier S ∩ S = {} ⟷ 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) ⊆ frontier S"
  by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)

lemma closure_Un_frontier: "closure S = S ∪ frontier S"
  by (simp add: closure_def frontier_closures sup_inf_distrib1)


subsection✐‹tag unimportant› ‹Filters and the ``eventually true'' quantifier›

text ‹Identify Trivial limits, where we can't approach arbitrarily closely.›

lemma trivial_limit_within: "trivial_limit (at a within S) ⟷ ¬ a islimpt S"
    unfolding trivial_limit_def eventually_at_topological islimpt_def
    by blast

lemma trivial_limit_at_iff: "trivial_limit (at a) ⟷ ¬ a islimpt UNIV"
  using trivial_limit_within [of a UNIV] by simp

lemma trivial_limit_at: "¬ trivial_limit (at a)"
  for a :: "'a::perfect_space"
  by (rule at_neq_bot)

lemma not_trivial_limit_within: "¬ trivial_limit (at x within S) = (x ∈ closure (S - {x}))"
  using islimpt_in_closure by (metis trivial_limit_within)

lemma not_in_closure_trivial_limitI:
  "x ∉ closure S ⟹ 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 ∈ closure S ⟹ filterlim f l (at x within S)"
  by (metis bot.extremum filterlim_iff_le_filtercomap not_in_closure_trivial_limitI that)

lemma at_within_eq_bot_iff: "at c within A = bot ⟷ c ∉ closure (A - {c})"
  using not_trivial_limit_within[of c A] by blast

subsection ‹Limits›

text ‹The expected monotonicity property.›

lemma Lim_Un:
  assumes "(f ⤏ l) (at x within S)" "(f ⤏ l) (at x within T)"
  shows "(f ⤏ l) (at x within (S ∪ T))"
  using assms unfolding at_within_union by (rule filterlim_sup)

lemma Lim_Un_univ:
  "(f ⤏ l) (at x within S) ⟹ (f ⤏ l) (at x within T) ⟹
    S ∪ T = UNIV ⟹ (f ⤏ l) (at x)"
  by (metis Lim_Un)

text ‹Interrelations between restricted and unrestricted limits.›

lemma Lim_at_imp_Lim_at_within: "(f ⤏ l) (at x) ⟹ (f ⤏ l) (at x within S)"
  by (metis order_refl filterlim_mono subset_UNIV at_le)

lemma eventually_within_interior:
  assumes "x ∈ interior S"
  shows "eventually P (at x within S) ⟷ eventually P (at x)"
  by (metis assms at_within_open_subset interior_subset open_interior)

lemma at_within_interior: "x ∈ interior S ⟹ 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 "∀S. (∀n. S n ≠ a ∧ S n ∈ T) ∧ S ⇢ a ⟶ (λn. X (S n)) ⇢ L"
  shows "(X ⤏ 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} ⇒
    'b::{linorder_topology, conditionally_complete_linorder}"
  assumes mono: "⋀a b. a ∈ I ⟹ b ∈ I ⟹ x < a ⟹ a ≤ b ⟹ f a ≤ f b"
    and bnd: "⋀a. a ∈ I ⟹ x < a ⟹ K ≤ f a"
  shows "(f ⤏ Inf (f ` ({x<..} ∩ I))) (at x within ({x<..} ∩ I))"
proof (cases "{x<..} ∩ 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<..} ∩ I))"
    {
      fix y
      assume "y ∈ {x<..} ∩ I"
      with False bnd have "Inf (f ` ({x<..} ∩ I)) ≤ f y"
        by (auto intro!: cInf_lower bdd_belowI2)
      with a have "a < f y"
        by (blast intro: less_le_trans)
    }
    then show "eventually (λx. a < f x) (at x within ({x<..} ∩ I))"
      by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
  next
    fix a
    assume "Inf (f ` ({x<..} ∩ I)) < a"
    from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y ∈ I" "f y < a"
      by auto
    then have "eventually (λx. x ∈ I ⟶ f x < a) (at_right x)"
      unfolding eventually_at_right[OF ‹x < y›] by (metis less_imp_le le_less_trans mono)
    then show "eventually (λx. f x < a) (at x within ({x<..} ∩ I))"
      unfolding eventually_at_filter by eventually_elim simp
  qed
qed

text‹These are special for limits out of the same topological space.›

lemma Lim_within_id: "(id ⤏ a) (at a within s)"
  unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_id: "(id ⤏ a) (at a)"
  unfolding id_def by (rule tendsto_ident_at)

text‹It's also sometimes useful to extract the limit point from the filter.›

abbreviation netlimit :: "'a::t2_space filter ⇒ 'a"
  where "netlimit F ≡ Lim F (λx. 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 ∈ interior S ⟹ (f ⤏ l) (at x within S) ⟷ (f ⤏ l) (at x)"
  by (metis at_within_interior)

lemma netlimit_within_interior:
  fixes x :: "'a::{t2_space,perfect_space}"
  assumes "x ∈ interior S"
  shows "netlimit (at x within S) = x"
  using assms by (metis at_within_interior netlimit_at)

text‹Useful lemmas on closure and set of possible sequential limits.›

lemma closure_sequential:
  fixes l :: "'a::first_countable_topology"
  shows "l ∈ closure S ⟷ (∃x. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially)"
  by (metis Diff_empty Diff_insert0 Un_iff closure_def islimpt_sequential mem_Collect_eq tendsto_const)

lemma closed_sequential_limits:
  fixes S :: "'a::first_countable_topology set"
  shows "closed S ⟷ (∀x l. (∀n. x n ∈ S) ∧ (x ⤏ l) sequentially ⟶ l ∈ S)"
by (metis closure_sequential closure_subset_eq subset_iff)

lemma tendsto_If_within_closures:
  assumes f: "x ∈ S ∪ (closure S ∩ closure T) ⟹
      (f ⤏ l x) (at x within S ∪ (closure S ∩ closure T))"
  assumes g: "x ∈ T ∪ (closure S ∩ closure T) ⟹
      (g ⤏ l x) (at x within T ∪ (closure S ∩ closure T))"
  assumes "x ∈ S ∪ T"
  shows "((λx. if x ∈ S then f x else g x) ⤏ l x) (at x within S ∪ T)"
proof -
  have *: "(S ∪ T) ∩ {x. x ∈ S} = S" "(S ∪ T) ∩ {x. x ∉ S} = T - S"
    by auto
  have "(f ⤏ l x) (at x within S)"
    by (rule filterlim_at_within_closure_implies_filterlim)
       (use ‹x ∈ _› in ‹auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]›)
  moreover
  have "(g ⤏ l x) (at x within T - S)"
    by (rule filterlim_at_within_closure_implies_filterlim)
      (use ‹x ∈ _› in
        ‹auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset›)
  ultimately show ?thesis
    by (intro filterlim_at_within_If) (simp_all only: *)
qed


subsection ‹Compactness›

lemma brouwer_compactness_lemma:
  fixes f :: "'a::topological_space ⇒ 'b::real_normed_vector"
  assumes "compact S"
    and "continuous_on S f"
    and "¬ (∃x∈S. f x = 0)"
  obtains d where "0 < d" and "∀x∈S. d ≤ 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 ∘ f)"
    by (rule continuous_intros continuous_on_norm assms(2))+
  with False obtain x where x: "x ∈ S" "∀y∈S. (norm ∘ f) x ≤ (norm ∘ f) y"
    using continuous_attains_inf[OF assms(1), of "norm ∘ f"]
    unfolding o_def
    by auto
  then show ?thesis
    by (metis assms(3) that comp_apply zero_less_norm_iff)
qed

subsubsection ‹Bolzano-Weierstrass property›

proposition Heine_Borel_imp_Bolzano_Weierstrass:
  assumes "compact S"
    and "infinite T"
    and "T ⊆ S"
  shows "∃x ∈ S. x islimpt T"
proof (rule ccontr)
  assume "¬ (∃x ∈ S. x islimpt T)"
  then obtain f where f: "∀x∈S. x ∈ f x ∧ open (f x) ∧ (∀y∈T. y ∈ f x ⟶ y = x)"
    unfolding islimpt_def by metis
  obtain g where g: "g ⊆ {T. ∃x. x ∈ S ∧ T = f x}" "finite g" "S ⊆ ⋃g"
    using assms(1)[unfolded compact_eq_Heine_Borel, THEN spec[where x="{T. ∃x. x∈S ∧ T = f x}"]]
    using f by auto
  then have g': "∀x∈g. ∃y ∈ S. x = f y"
    by auto
  have "inj_on f T"
    by (smt (verit, best) assms(3) f inj_onI subsetD)
  then have "infinite (f ` T)"
    using assms(2) using finite_imageD by auto
  moreover
    have False if "x ∈ T" "f x ∉ g" for x
      by (smt (verit) UnionE assms(3) f g' g(3) subsetD that)
  then have "f ` T ⊆ g" by auto
  ultimately show False
    using g(2) using finite_subset by auto
qed

lemma sequence_infinite_lemma:
  fixes f :: "nat ⇒ 'a::t1_space"
  assumes "∀n. f n ≠ l"
    and "(f ⤏ l) sequentially"
  shows "infinite (range f)"
proof
  assume "finite (range f)"
  then have "l ∉ range f ∧ closed (range f)"
    using ‹finite (range f)› assms(1) finite_imp_closed by blast
  then have "eventually (λn. f n ∈ - range f) sequentially"
    by (metis Compl_iff assms(2) open_Compl 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 "∀T. infinite T ∧ T ⊆ S --> (∃x ∈ S. x islimpt T)"
  shows "closed S"
proof -
  {
    fix x l
    assume as: "∀n::nat. x n ∈ S" "(x ⤏ l) sequentially"
    have "l ∈ S"
    proof (cases "∀n. x n ≠ l")
      case False
      then show "l∈S" using as(1) by auto
    next
      case True
      with as(2) have "infinite (range x)"
        using sequence_infinite_lemma[of x l] by auto
      then obtain l' where "l'∈S" "l' islimpt (range x)"
        using as(1) assms by auto
      then show "l∈S" using sequence_unique_limpt as True 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)"
  by (metis closed_singleton closure_Un closure_closed insert_is_Un)

lemma finite_not_islimpt_in_compact:
  assumes "compact A" "⋀z. z ∈ A ⟹ ¬z islimpt B"
  shows   "finite (A ∩ B)"
  by (meson Heine_Borel_imp_Bolzano_Weierstrass assms inf_le1 inf_le2 islimpt_subset)


text‹In particular, some common special cases.›

lemma compact_Un [intro]:
  assumes "compact S"
    and "compact T"
  shows " compact (S ∪ T)"
proof (rule compactI)
  fix f
  assume *: "Ball f open" "S ∪ T ⊆ ⋃f"
  from * ‹compact S› obtain s' where "s' ⊆ f ∧ finite s' ∧ S ⊆ ⋃s'"
    unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
  moreover
  from * ‹compact T› obtain t' where "t' ⊆ f ∧ finite t' ∧ T ⊆ ⋃t'"
    unfolding compact_eq_Heine_Borel by (auto elim!: allE[of _ f])
  ultimately show "∃f'⊆f. finite f' ∧ S ∪ T ⊆ ⋃f'"
    by (auto intro!: exI[of _ "s' ∪ t'"])
qed

lemma compact_Union [intro]: "finite S ⟹ (⋀T. T ∈ S ⟹ compact T) ⟹ compact (⋃S)"
  by (induct set: finite) auto

lemma compact_UN [intro]:
  "finite A ⟹ (⋀x. x ∈ A ⟹ compact (B x)) ⟹ compact (⋃x∈A. B x)"
  by blast

lemma closed_Int_compact [intro]:
  assumes "closed S" and "compact T"
  shows "compact (S ∩ 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 ∩ 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)"
  by (metis assms compact_Un compact_sing insert_is_Un)

lemma finite_imp_compact: "finite S ⟹ compact S"
  by (induct set: finite) simp_all

lemma open_delete:
  fixes S :: "'a::t1_space set"
  shows "open S ⟹ open (S - {x})"
  by (simp add: open_Diff)


text‹Compactness expressed with filters›

lemma closure_iff_nhds_not_empty:
  "x ∈ closure X ⟷ (∀A. ∀S⊆A. open S ⟶ x ∈ S ⟶ X ∩ A ≠ {})"
proof safe
  assume x: "x ∈ closure X"
  fix S A
  assume §: "open S" "x ∈ S" "X ∩ A = {}" "S ⊆ A"
  then have "x ∉ closure (-S)"
    by (simp add: closed_open)
  with x have "x ∈ closure X - closure (-S)"
    by auto
  with § show False
    by (metis Compl_iff Diff_iff closure_mono disjoint_iff subsetD subsetI)
next
  assume "∀A S. S ⊆ A ⟶ open S ⟶ x ∈ S ⟶ X ∩ A ≠ {}"
  then show "x ∈ closure X"
    by (metis ComplI Compl_disjoint closure_interior interior_subset open_interior)
qed

lemma compact_filter:
  "compact U ⟷ (∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
  fix F
  assume "compact U"
  assume F: "F ≠ bot" "eventually (λx. x ∈ U) F"
  then have "U ≠ {}"
    by (auto simp: eventually_False)

  define Z where "Z = closure ` {A. eventually (λx. x ∈ A) F}"
  then have "∀z∈Z. closed z"
    by auto
  moreover
  have ev_Z: "⋀z. z ∈ Z ⟹ eventually (λx. x ∈ z) F"
    unfolding Z_def by (auto elim: eventually_mono intro: subsetD[OF closure_subset])
  have "(∀B ⊆ Z. finite B ⟶ U ∩ ⋂B ≠ {})"
  proof (intro allI impI)
    fix B assume "finite B" "B ⊆ Z"
    with ‹finite B› ev_Z F(2) have "eventually (λx. x ∈ U ∩ (⋂B)) F"
      by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
    with F show "U ∩ ⋂B ≠ {}"
      by (intro notI) (simp add: eventually_False)
  qed
  ultimately have "U ∩ ⋂Z ≠ {}"
    using ‹compact U› unfolding compact_fip by blast
  then obtain x where "x ∈ U" and x: "⋀z. z ∈ Z ⟹ x ∈ z"
    by auto

  have "⋀P. eventually P (inf (nhds x) F) ⟹ P ≠ bot"
    unfolding eventually_inf eventually_nhds
  proof safe
    fix P Q R S
    assume "eventually R F" "open S" "x ∈ S"
    with open_Int_closure_eq_empty[of S "{x. R x}"] x
    have "S ∩ {x. R x} ≠ {}" by (auto simp: Z_def)
    moreover assume "Ball S Q" "∀x. Q x ∧ R x ⟶ bot x"
    ultimately show False by (auto simp: set_eq_iff)
  qed
  with ‹x ∈ U› show "∃x∈U. inf (nhds x) F ≠ bot"
    by (metis eventually_bot)
next
  fix A
  assume A: "∀a∈A. closed a" "∀B⊆A. finite B ⟶ U ∩ ⋂B ≠ {}" "U ∩ ⋂A = {}"
  define F where "F = (INF a∈insert U A. principal a)"
  have "F ≠ bot"
    unfolding F_def
  proof (rule INF_filter_not_bot)
    fix X
    assume X: "X ⊆ insert U A" "finite X"
    with A(2)[THEN spec, of "X - {U}"] have "U ∩ ⋂(X - {U}) ≠ {}"
      by auto
    with X show "(INF a∈X. principal a) ≠ bot"
      by (auto simp: INF_principal_finite principal_eq_bot_iff)
  qed
  moreover
  have "F ≤ principal U"
    unfolding F_def by auto
  then have "eventually (λx. x ∈ U) F"
    by (auto simp: le_filter_def eventually_principal)
  moreover
  assume "∀F. F ≠ bot ⟶ eventually (λx. x ∈ U) F ⟶ (∃x∈U. inf (nhds x) F ≠ bot)"
  ultimately obtain x where "x ∈ U" and x: "inf (nhds x) F ≠ bot"
    by auto

  { fix V assume "V ∈ A"
    then have "F ≤ principal V"
      unfolding F_def by (intro INF_lower2[of V]) auto
    then have V: "eventually (λx. x ∈ V) F"
      by (auto simp: le_filter_def eventually_principal)
    have "x ∈ closure V"
      unfolding closure_iff_nhds_not_empty
    proof (intro impI allI)
      fix S A
      assume "open S" "x ∈ S" "S ⊆ A"
      then have "eventually (λx. x ∈ A) (nhds x)"
        by (auto simp: eventually_nhds)
      with V have "eventually (λx. x ∈ V ∩ A) (inf (nhds x) F)"
        by (auto simp: eventually_inf)
      with x show "V ∩ A ≠ {}"
        by (auto simp del: Int_iff simp add: trivial_limit_def)
    qed
    then have "x ∈ V"
      using ‹V ∈ A› A(1) by simp
  }
  with ‹x∈U› have "x ∈ U ∩ ⋂A" by auto
  with ‹U ∩ ⋂A = {}› show False by auto
qed

definition✐‹tag important› countably_compact :: "('a::topological_space) set ⇒ bool" where
"countably_compact U ⟷
  (∀A. countable A ⟶ (∀a∈A. open a) ⟶ U ⊆ ⋃A
     ⟶ (∃T⊆A. finite T ∧ U ⊆ ⋃T))"

lemma countably_compactE:
  assumes "countably_compact s" and "∀t∈C. open t" and "s ⊆ ⋃C" "countable C"
  obtains C' where "C' ⊆ C" and "finite C'" and "s ⊆ ⋃C'"
  using assms unfolding countably_compact_def by metis

lemma countably_compactI:
  assumes "⋀C. ∀t∈C. open t ⟹ s ⊆ ⋃C ⟹ countable C ⟹ (∃C'⊆C. finite C' ∧ s ⊆ ⋃C')"
  shows "countably_compact s"
  using assms unfolding countably_compact_def by metis

lemma compact_imp_countably_compact: "compact U ⟹ 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" "∀b∈B. open b"
    and basis: "⋀T x. open T ⟹ x ∈ T ⟹ x ∈ U ⟹ ∃b∈B. x ∈ b ∧ b ∩ U ⊆ T"
  shows "compact U"
  using ‹countably_compact U›
  unfolding compact_eq_Heine_Borel countably_compact_def
proof safe
  fix A
  assume A: "∀a∈A. open a" "U ⊆ ⋃A"
  assume *: "∀A. countable A ⟶ (∀a∈A. open a) ⟶ U ⊆ ⋃A ⟶ (∃T⊆A. finite T ∧ U ⊆ ⋃T)"
  moreover define C where "C = {b∈B. ∃a∈A. b ∩ U ⊆ a}"
  ultimately have "countable C" "∀a∈C. open a"
    unfolding C_def using ccover by auto
  moreover
  have "⋃A ∩ U ⊆ ⋃C"
  proof safe
    fix x a
    assume "x ∈ U" "x ∈ a" "a ∈ A"
    with basis[of a x] A obtain b where "b ∈ B" "x ∈ b" "b ∩ U ⊆ a"
      by blast
    with ‹a ∈ A› show "x ∈ ⋃C"
      unfolding C_def by auto
  qed
  then have "U ⊆ ⋃C" using ‹U ⊆ ⋃A› by auto
  ultimately obtain T where T: "T⊆C" "finite T" "U ⊆ ⋃T"
    using * by metis
  then have "∀t∈T. ∃a∈A. t ∩ U ⊆ a"
    by (auto simp: C_def)
  then obtain f where "∀t∈T. f t ∈ A ∧ t ∩ U ⊆ f t"
    unfolding bchoice_iff Bex_def ..
  with T show "∃T⊆A. finite T ∧ U ⊆ ⋃T"
    unfolding C_def by (intro exI[of _ "f`T"]) fastforce
qed

proposition countably_compact_imp_compact_second_countable:
  "countably_compact U ⟹ compact (U :: 'a :: second_countable_topology set)"
proof (rule countably_compact_imp_compact)
  fix T and x :: 'a
  assume "open T" "x ∈ T"
  from topological_basisE[OF is_basis this] obtain b where
    "b ∈ (SOME B. countable B ∧ topological_basis B)" "x ∈ b" "b ⊆ T" .
  then show "∃b∈SOME B. countable B ∧ topological_basis B. x ∈ b ∧ b ∩ U ⊆ T"
    by blast
qed (insert countable_basis topological_basis_open[OF is_basis], auto)

lemma countably_compact_eq_compact:
  "countably_compact U ⟷ compact (U :: 'a :: second_countable_topology set)"
  using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

subsubsection‹Sequential compactness›

definition✐‹tag important› seq_compact :: "'a::topological_space set ⇒ bool" where
  "seq_compact S ⟷
    (∀f. (∀n. f n ∈ S) ⟶ (∃l∈S. ∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l))"

lemma seq_compactI:
  assumes "⋀f. ∀n. f n ∈ S ⟹ ∃l∈S. ∃r::nat⇒nat. strict_mono r ∧ (f ∘ r) ⇢ l"
  shows "seq_compact S"
  unfolding seq_compact_def using assms by fast

lemma seq_compactE:
  assumes "seq_compact S" "∀n. f n ∈ S"
  obtains l r where "l ∈ S" "strict_mono (r :: nat ⇒ nat)" "(f ∘ r) ⇢ l"
  using assms unfolding seq_compact_def by fast

lemma seq_compact_Int_closed:
  assumes "seq_compact S" and "closed T"
  shows "seq_compact (S ∩ T)"
proof (rule seq_compactI)
  fix f assume "∀n::nat. f n ∈ S ∩ T"
  hence "∀n. f n ∈ S" and "∀n. f n ∈ T"
    by simp_all
  from ‹seq_compact S› and ‹∀n. f n ∈ S›
  obtain l r where "l ∈ S" and r: "strict_mono r" and l: "(f ∘ r) ⇢ l"
    by (rule seq_compactE)
  from ‹∀n. f n ∈ T› have "∀n. (f ∘ r) n ∈ T"
    by simp
  with ‹l ∈ S› and r and l show "∃l∈S ∩ T. ∃r. strict_mono r ∧ (f ∘ r) ⇢ l"
    by (metis Int_iff ‹closed T› closed_sequentially)
qed

lemma seq_compact_closed_subset:
  assumes "closed S" and "S ⊆ 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: "∀a∈A. open a" "U ⊆ ⋃A" "countable A"
  have subseq: "⋀X. range X ⊆ U ⟹ ∃r x. x ∈ U ∧ strict_mono (r :: nat ⇒ nat) ∧ (X ∘ r) ⇢ x"
    using ‹seq_compact U› by (fastforce simp: seq_compact_def subset_eq)
  show "∃T⊆A. finite T ∧ U ⊆ ⋃T"
  proof cases
    assume "finite A"
    with A show ?thesis by auto
  next
    assume "infinite A"
    then have "A ≠ {}" by auto
    show ?thesis
    proof (rule ccontr)
      assume "¬ (∃T⊆A. finite T ∧ U ⊆ ⋃T)"
      then have "∀T. ∃x. T ⊆ A ∧ finite T ⟶ (x ∈ U - ⋃T)"
        by auto
      then obtain X' where T: "⋀T. T ⊆ A ⟹ finite T ⟹ X' T ∈ U - ⋃T"
        by metis
      define X where "X n = X' (from_nat_into A ` {.. n})" for n
      have X: "⋀n. X n ∈ U - (⋃i≤n. from_nat_into A i)"
        using ‹A ≠ {}› unfolding X_def by (intro T) (auto intro: from_nat_into)
      then have "range X ⊆ U"
        by auto
      with subseq[of X] obtain r x where "x ∈ U" and r: "strict_mono r" "(X ∘ r) ⇢ x"
        by auto
      from ‹x∈U› ‹U ⊆ ⋃A› from_nat_into_surj[OF ‹countable A›]
      obtain n where "x ∈ from_nat_into A n" by auto
      with r(2) A(1) from_nat_into[OF ‹A ≠ {}›]
      have "eventually (λi. X (r i) ∈ from_nat_into A n) sequentially"
        unfolding tendsto_def by fastforce
      then obtain N where "⋀i. N ≤ i ⟹ X (r i) ∈ from_nat_into A n"
        by (auto simp: eventually_sequentially)
      moreover from X have "⋀i. n ≤ r i ⟹ X (r i) ∉ from_nat_into A n"
        by auto
      moreover from ‹strict_mono r›[THEN seq_suble, of "max n N"] have "∃i. n ≤ r i ∧ N ≤ 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 ⇒ 'a"
  assume "∀n. X n ∈ U"
  then have "eventually (λx. x ∈ U) (filtermap X sequentially)"
    by (auto simp: eventually_filtermap)
  moreover
  have "filtermap X sequentially ≠ bot"
    by (simp add: trivial_limit_def eventually_filtermap)
  ultimately
  obtain x where "x ∈ U" and x: "inf (nhds x) (filtermap X sequentially) ≠ bot" (is "?F ≠ _")
    using ‹compact U› by (auto simp: compact_filter)

  from countable_basis_at_decseq[of x]
  obtain A where A:
      "⋀i. open (A i)"
      "⋀i. x ∈ A i"
      "⋀S. open S ⟹ x ∈ S ⟹ eventually (λi. A i ⊆ S) sequentially"
    by blast
  define s where "s n i = (SOME j. i < j ∧ X j ∈ A (Suc n))" for n i
  {
    fix n i
    have "∃a. i < a ∧ X a ∈ A (Suc n)"
    proof (rule ccontr)
      assume "¬ (∃a>i. X a ∈ A (Suc n))"
      then have "⋀a. Suc i ≤ a ⟹ X a ∉ A (Suc n)"
        by auto
      then have "eventually (λx. x ∉ A (Suc n)) (filtermap X sequentially)"
        by (auto simp: eventually_filtermap eventually_sequentially)
      moreover have "eventually (λx. x ∈ A (Suc n)) (nhds x)"
        using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
      ultimately have "eventually (λx. 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) ∈ 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 "(λn. X (r n)) ⇢ x"
  proof (rule topological_tendstoI)
    fix S
    assume "open S" "x ∈ S"
    with A(3) have "eventually (λi. A i ⊆ S) sequentially"
      by auto
    moreover
    {
      fix i
      assume "Suc 0 ≤ i"
      then have "X (r i) ∈ A i"
        by (cases i) (simp_all add: r_def s)
    }
    then have "eventually (λi. X (r i) ∈ A i) sequentially"
      by (auto simp: eventually_sequentially)
    ultimately show "eventually (λi. X (r i) ∈ S) sequentially"
      by eventually_elim auto
  qed
  ultimately show "∃x ∈ U. ∃r. strict_mono r ∧ (X ∘ r) ⇢ x"
    using ‹x ∈ U› 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 ⊆ S"
  shows "∃x∈S. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ T)"
proof (rule ccontr)
  define C where "C = (λF. interior (F ∪ (- T))) ` {F. finite F ∧ F ⊆ T }"
  note ‹countably_compact S›
  moreover have "∀T∈C. open T"
    by (auto simp: C_def)
  moreover
  assume "¬ (∃x∈S. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ T))"
  then have S: "⋀x. x ∈ S ⟹ ∃U. x∈U ∧ open U ∧ finite (U ∩ T)" by metis
  have "S ⊆ ⋃C"
    using ‹T ⊆ S›
    unfolding C_def
    apply (safe dest!: S)
    apply (rule_tac a="U ∩ T" in UN_I)
    apply (auto intro!: interiorI simp add: finite_subset)
    done
  moreover
  from ‹countable T› have "countable C"
    unfolding C_def by (auto intro: countable_Collect_finite_subset)
  ultimately
  obtain D where "D ⊆ C" "finite D" "S ⊆ ⋃D"
    by (rule countably_compactE)
  then obtain E where E: "E ⊆ {F. finite F ∧ F ⊆ T }" "finite E"
    and S: "S ⊆ (⋃F∈E. interior (F ∪ (- T)))"
    by (metis (lifting) finite_subset_image C_def)
  from S ‹T ⊆ S› have "T ⊆ ⋃E"
    using interior_subset by blast
  moreover have "finite (⋃E)"
    using E by auto
  ultimately show False using ‹infinite T›
    by (auto simp: finite_subset)
qed

lemma countable_acc_point_imp_seq_compact:
  fixes S :: "'a::first_countable_topology set"
  assumes "⋀T. ⟦infinite T; countable T; T ⊆ S⟧ ⟹ ∃x∈S. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ T)"
  shows "seq_compact S"
  unfolding seq_compact_def
proof (intro strip)
  fix f :: "nat ⇒ 'a"
  assume f: "∀n. f n ∈ S"
  show "∃l∈S. ∃r. strict_mono r ∧ ((f ∘ r) ⤏ 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 ⇒ nat" where "strict_mono  r" and fr: "∀n. f (r n) = f l"
      using infinite_enumerate by blast
    then have "strict_mono r ∧ (f ∘ r) ⇢ f l"
      by (simp add: fr o_def)
    with f show "∃l∈S. ∃r. strict_mono  r ∧ (f ∘ r) ⇢ l"
      by auto
  next
    case False
    with f assms obtain l where "l ∈ S" "∀U. l∈U ∧ open U ⟶ infinite (U ∩ range f)"
      by (metis image_subset_iff uncountable_def) 
    with ‹l ∈ S› show "∃l∈S. ∃r. strict_mono r ∧ ((f ∘ r) ⤏ l) sequentially"
      by (meson acc_point_range_imp_convergent_subsequence) 
  qed
qed

lemma seq_compact_eq_countably_compact:
  fixes U :: "'a :: first_countable_topology set"
  shows "seq_compact U ⟷ countably_compact U"
  by (metis countable_acc_point_imp_seq_compact countably_compact_imp_acc_point seq_compact_imp_countably_compact)

lemma seq_compact_eq_acc_point:
  fixes S :: "'a :: first_countable_topology set"
  shows "seq_compact S ⟷
    (∀T. infinite T ∧ countable T ∧ T ⊆ S --> (∃x∈S. ∀U. x∈U ∧ open U ⟶ infinite (U ∩ T)))"
  by (metis countable_acc_point_imp_seq_compact countably_compact_imp_acc_point seq_compact_imp_countably_compact)

lemma seq_compact_eq_compact:
  fixes U :: "'a :: second_countable_topology set"
  shows "seq_compact U ⟷ 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 "(⋀T. ⟦infinite T; T ⊆ S⟧ ⟹∃x ∈ S. x islimpt T) ⟹ seq_compact S"
  by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)


subsection✐‹tag unimportant› ‹Cartesian products›

lemma seq_compact_Times:
  assumes "seq_compact S" "seq_compact T"
  shows "seq_compact (S × T)"
  unfolding seq_compact_def
proof clarify
  fix h :: "nat ⇒ 'a × 'b"
  assume "∀n. h n ∈ S × T"
  then have *: "⋀n. (fst ∘ h) n ∈ S" "⋀n. (snd ∘ h) n ∈ T"
    by (simp_all add: mem_Times_iff)
  then obtain lS and rS :: "nat⇒nat"
    where "lS∈S" "strict_mono rS" and lS: "(fst ∘ h ∘ rS) ⇢ lS"
    using assms seq_compact_def by metis
  then obtain lT and rT :: "nat⇒nat" 
    where  "lT∈T" "strict_mono rT" and lT: "(snd ∘ h ∘ rS ∘ rT) ⇢ lT"
    using assms seq_compact_def *
    by (metis (mono_tags, lifting) comp_apply) 
  have "strict_mono (rS ∘ rT)"
    by (simp add: ‹strict_mono rS› ‹strict_mono rT› strict_mono_o)
  moreover have "(h ∘ (rS ∘ rT)) ⇢ (lS,lT)"
    using tendsto_Pair [OF LIMSEQ_subseq_LIMSEQ [OF lS ‹strict_mono rT›] lT]
    by (simp add: o_def)
  ultimately show "∃l∈S × T. ∃r. strict_mono r ∧ (h ∘ r) ⇢ l"
    using ‹lS ∈ S› ‹lT ∈ T› by blast
qed

lemma compact_Times:
  assumes "compact S" "compact T"
  shows "compact (S × T)"
proof (rule compactI)
  fix 𝒞
  assume C: "∀T∈𝒞. open T" "S × T ⊆ ⋃𝒞"
  have "∀x∈S. ∃A. open A ∧ x ∈ A ∧ (∃D⊆𝒞. finite D ∧ A × T ⊆ ⋃D)"
  proof
    fix x
    assume "x ∈ S"
    have "∀y∈T. ∃A B C. C ∈ 𝒞 ∧ open A ∧ open B ∧ x ∈ A ∧ y ∈ B ∧ A × B ⊆ C"
      by (smt (verit, ccfv_threshold) C UnionE ‹x ∈ S› mem_Sigma_iff open_prod_def subsetD)
    then obtain a b c where b: "⋀y. y ∈ T ⟹ open (b y)"
      and c: "⋀y. y ∈ T ⟹ c y ∈ 𝒞 ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y"
      by metis
    then have "∀y∈T. open (b y)" "T ⊆ (⋃y∈T. b y)" by auto
    with compactE_image[OF ‹compact T›] obtain D where D: "D ⊆ T" "finite D" "T ⊆ (⋃y∈D. b y)"
      by metis
    moreover from D c have "(⋂y∈D. a y) × T ⊆ (⋃y∈D. c y)"
      by (fastforce simp: subset_eq)
    ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆𝒞. finite d ∧ a × T ⊆ ⋃d)"
      using c by (intro exI[of _ "c`D"] exI[of _ "⋂(a`D)"] conjI) (auto intro!: open_INT)
  qed
  then obtain a d where a: "⋀x. x∈S ⟹ open (a x)" "S ⊆ (⋃x∈S. a x)"
    and d: "⋀x. x ∈ S ⟹ d x ⊆ 𝒞 ∧ finite (d x) ∧ a x × T ⊆ ⋃(d x)"
    unfolding subset_eq UN_iff by metis
  moreover
  from compactE_image[OF ‹compact S› a]
  obtain e where e: "e ⊆ S" "finite e" and S: "S ⊆ (⋃x∈e. a x)"
    by auto
  moreover
  have "S × T ⊆ (⋃x∈e. ⋃(d x))"
    by (smt (verit, del_insts) S SigmaE UN_iff d e(1) mem_Sigma_iff subset_eq)
  ultimately show "∃C'⊆𝒞. finite C' ∧ S × T ⊆ ⋃C'"
    by (intro exI[of _ "(⋃x∈e. d x)"]) (auto simp: subset_eq)
qed


lemma tube_lemma:
  assumes "compact K"
  assumes "open W"
  assumes "{x0} × K ⊆ W"
  shows "∃X0. x0 ∈ X0 ∧ open X0 ∧ X0 × K ⊆ W"
proof -
  {
    fix y assume "y ∈ K"
    then have "(x0, y) ∈ W" using assms by auto
    with ‹open W›
    have "∃X0 Y. open X0 ∧ open Y ∧ x0 ∈ X0 ∧ y ∈ Y ∧ X0 × Y ⊆ W"
      by (rule open_prod_elim) blast
  }
  then obtain X0 Y where
    *: "∀y ∈ K. open (X0 y) ∧ open (Y y) ∧ x0 ∈ X0 y ∧ y ∈ Y y ∧ X0 y × Y y ⊆ W"
    by metis
  from * have "∀t∈Y ` K. open t" "K ⊆ ⋃(Y ` K)" by auto
  with ‹compact K› obtain CC where CC: "CC ⊆ Y ` K" "finite CC" "K ⊆ ⋃CC"
    by (meson compactE)
  then obtain c where c: "⋀C. C ∈ CC ⟹ c C ∈ K ∧ C = Y (c C)"
    by (force intro!: choice)
  with * CC show ?thesis
    by (force intro!: exI[where x="⋂C∈CC. X0 (c C)"]) (* SLOW *)
qed

lemma continuous_on_prod_compactE:
  fixes fx::"'a::topological_space × 'b::topological_space ⇒ 'c::metric_space"
    and e::real
  assumes cont_fx: "continuous_on (U × C) fx"
  assumes "compact C"
  assumes [intro]: "x0 ∈ U"
  notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
  assumes "e > 0"
  obtains X0 where "x0 ∈ X0" "open X0"
    "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof -
  define psi where "psi = (λ(x, t). dist (fx (x, t)) (fx (x0, t)))"
  define W0 where "W0 = {(x, t) ∈ U × C. psi (x, t) < e}"
  have W0_eq: "W0 = psi -` {..<e} ∩ U × C"
    by (auto simp: vimage_def W0_def)
  have "open {..<e}" by simp
  have "continuous_on (U × C) psi"
    by (auto intro!: continuous_intros simp: psi_def split_beta')
  then obtain W where W: "open W" "W ∩ U × C = W0 ∩ U × C"
    unfolding W0_eq
    by (metis ‹open {..<e}› continuous_on_open_invariant inf_right_idem) 
  have "{x0} × C ⊆ W ∩ U × C"
    unfolding W
    by (auto simp: W0_def psi_def ‹0 < e›)
  then have "{x0} × C ⊆ W" by blast
  from tube_lemma[OF ‹compact C› ‹open W› this]
  obtain X0 where X0: "x0 ∈ X0" "open X0" "X0 × C ⊆ W"
    by blast

  have "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
  proof safe
    fix x assume x: "x ∈ X0" "x ∈ U"
    fix t assume t: "t ∈ C"
    have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
      by (auto simp: psi_def)
    also have "psi (x, t) < e"
      using W(2) W0_def X0(3) t x by fastforce
    finally show "dist (fx (x,t)) (fx (x0,t)) ≤ e" by simp
  qed
  from X0(1,2) this show ?thesis ..
qed


subsection ‹Continuity›

lemma continuous_at_imp_continuous_within:
  "continuous (at x) f ⟹ 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 ⟹ (f ⤏ l) net"
  by simp

lemmas continuous_on = continuous_on_def ― ‹legacy theorem name›

lemma continuous_within_subset:
  "continuous (at x within S) f ⟹ t ⊆ S ⟹ continuous (at x within t) f"
  unfolding continuous_within by(metis tendsto_within_subset)

lemma continuous_on_interior:
  "continuous_on S f ⟹ x ∈ interior S ⟹ continuous (at x) f"
  by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)

lemma continuous_on_eq:
  "⟦continuous_on S f; ⋀x. x ∈ S ⟹ f x = g x⟧ ⟹ continuous_on S g"
  unfolding continuous_on_def tendsto_def eventually_at_topological
  by simp

text ‹Characterization of various kinds of continuity in terms of sequences.›

lemma continuous_within_sequentiallyI:
  fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
  assumes "⋀u::nat ⇒ 'a. u ⇢ a ⟹ (∀n. u n ∈ S) ⟹ (λn. f (u n)) ⇢ 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 ⇒ 'b::topological_space"
  assumes f: "continuous (at a within S) f"
         and  "eventually (λn. x n ∈ S) F" "(x ⤏ a) F "
  shows "((λn. f (x n)) ⤏ f a) F"
proof -
  have *: "filterlim x (inf (nhds a) (principal S)) F"
    by (simp add: assms filterlim_inf filterlim_principal)
  show ?thesis
    using "*" f continuous_within filterlim_compose tendsto_at_within_iff_tendsto_nhds by blast
qed

lemma continuous_within_tendsto_compose':
  fixes f::"'a::t2_space ⇒ 'b::topological_space"
  assumes "continuous (at a within S) f"
    "⋀n. x n ∈ S"
    "(x ⤏ a) F "
  shows "((λn. f (x n)) ⤏ f a) F"
  using always_eventually assms continuous_within_tendsto_compose by blast

lemma continuous_within_sequentially:
  fixes f :: "'a::{first_countable_topology, t2_space} ⇒ 'b::topological_space"
  shows "continuous (at a within S) f ⟷
    (∀x. (∀n::nat. x n ∈ S) ∧ (x ⤏ a) sequentially
         ⟶ ((f ∘ x) ⤏ 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} ⇒ 'b::topological_space"
  assumes "⋀u. u ⇢ a ⟹ (λn. f (u n)) ⇢ 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 ⇒ 'b::topological_space"
  shows "continuous (at a) f ⟷
    (∀x. (x ⤏ a) sequentially --> ((f ∘ x) ⤏ 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} ⇒ 'b::topological_space"
  assumes "⋀u a. (∀n. u n ∈ S) ⟹ a ∈ S ⟹ u ⇢ a ⟹ (λn. f (u n)) ⇢ 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} ⇒ 'b::topological_space"
  shows "continuous_on S f ⟷
    (∀x. ∀a ∈ S. (∀n. x(n) ∈ S) ∧ (x ⤏ a) sequentially
      ⟶ ((f ∘ x) ⤏ f a) sequentially)"
  by (meson continuous_on_eq_continuous_within continuous_within_sequentially)

text ‹Continuity in terms of open preimages.›

lemma continuous_at_open:
  "continuous (at x) f ⟷ (∀t. open t ∧ f x ∈ t ⟶ (∃S. open S ∧ x ∈ S ∧ (∀x' ∈ S. (f x') ∈ t)))"
  by (metis UNIV_I continuous_within_topological)

lemma continuous_imp_tendsto:
  assumes "continuous (at x0) f" and "x ⇢ x0"
  shows "(f ∘ x) ⇢ (f x0)"
proof (rule topological_tendstoI)
  fix S
  assume "open S" "f x0 ∈ S"
  then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S"
     using assms continuous_at_open by metis
  then have "eventually (λn. x n ∈ T) sequentially"
    using assms T_def by (auto simp: tendsto_def)
  then show "eventually (λn. (f ∘ x) n ∈ S) sequentially"
    using T_def by (auto elim!: eventually_mono)
qed

subsection ‹Homeomorphisms›

definition✐‹tag important› "homeomorphism S T f g ⟷
  (∀x∈S. (g(f x) = x)) ∧ (f ` S = T) ∧ continuous_on S f ∧
  (∀y∈T. (f(g y) = y)) ∧ (g ` T = S) ∧ continuous_on T g"

lemma homeomorphismI [intro?]:
  assumes "continuous_on S f" "continuous_on T g"
    "f ` S ⊆ T" "g ` T ⊆ S" "⋀x. x ∈ S ⟹ g(f x) = x" "⋀y. y ∈ T ⟹ 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 (λa. a) (λa. 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" "⋀x. x ∈ X ⟹ f' x = f x" "⋀y. y ∈ Y ⟹ 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 ⟹ 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)

lemma continuous_on_translation_eq:
  fixes g :: "'a :: real_normed_vector ⇒ 'b :: real_normed_vector"
  shows "continuous_on A ((+) a ∘ g) = continuous_on A g"
proof -
  have g: "g = (λx. -a + x) ∘ ((λx. a + x) ∘ g)"
    by (rule ext) simp
  show ?thesis
    by (metis (no_types, opaque_lifting) g continuous_on_compose homeomorphism_def homeomorphism_translation)
qed

definition✐‹tag important› homeomorphic :: "'a::topological_space set ⇒ 'b::topological_space set ⇒ bool"
    (infixr ‹homeomorphic› 60)
  where "s homeomorphic t ≡ (∃f g. homeomorphism s t f g)"

lemma homeomorphic_empty [iff]:
     "S homeomorphic {} ⟷ S = {}" "{} homeomorphic S ⟷ S = {}"
  by (auto simp: homeomorphic_def homeomorphism_def)

lemma homeomorphic_refl: "S homeomorphic S"
  using homeomorphic_def homeomorphism_ident by fastforce

lemma homeomorphic_sym: "S homeomorphic T ⟷ 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 ⟷
    (∃f g. (∀x∈S. f(x) ∈ T ∧ (g(f(x)) = x)) ∧
           (∀y∈T. g(y) ∈ S ∧ (f(g(y)) = y)) ∧
           continuous_on S f ∧ continuous_on T g)"
  by (smt (verit, ccfv_threshold) homeomorphic_def homeomorphismI homeomorphism_def image_eqI image_subset_iff)

lemma homeomorphicI [intro?]:
   "⟦f ` S = T; g ` T = S;
     continuous_on S f; continuous_on T g;
     ⋀x. x ∈ S ⟹ g(f(x)) = x;
     ⋀y. y ∈ T ⟹ f(g(y)) = y⟧ ⟹ S homeomorphic T"
unfolding homeomorphic_def homeomorphism_def by metis

lemma homeomorphism_of_subsets:
   "⟦homeomorphism S T f g; S' ⊆ S; T'' ⊆ T; f ` S' = T'⟧
    ⟹ homeomorphism S' T' f g"
  by (smt (verit, del_insts) continuous_on_subset homeomorphismI homeomorphism_def imageE subset_eq)

lemma homeomorphism_apply1: "⟦homeomorphism S T f g; x ∈ S⟧ ⟹ g(f x) = x"
  by (simp add: homeomorphism_def)

lemma homeomorphism_apply2: "⟦homeomorphism S T f g; x ∈ T⟧ ⟹ f(g x) = x"
  by (simp add: homeomorphism_def)

lemma homeomorphism_image1: "homeomorphism S T f g ⟹ f ` S = T"
  by (simp add: homeomorphism_def)

lemma homeomorphism_image2: "homeomorphism S T f g ⟹ g ` T = S"
  by (simp add: homeomorphism_def)

lemma homeomorphism_cont1: "homeomorphism S T f g ⟹ continuous_on S f"
  by (simp add: homeomorphism_def)

lemma homeomorphism_cont2: "homeomorphism S T f g ⟹ continuous_on T g"
  by (simp add: homeomorphism_def)

lemma continuous_on_no_limpt:
   "(⋀x. ¬ x islimpt S) ⟹ 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 ⟹ 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 ⟷ finite S ∧ finite T ∧ card S = card T" (is "?lhs = ?rhs")
proof
  assume "S homeomorphic T"
  with assms show ?rhs
    by (metis (full_types) card_image_le finite_imageI homeomorphic_def homeomorphism_def 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 (simp only: homeomorphic_def homeomorphism_def continuous_on_finite)
    by (smt (verit, ccfv_SIG) bij_betw_imp_surj_on bij_betw_inv_into bij_betw_inv_into_left bij_betw_inv_into_right)
qed

text ‹Relatively weak hypotheses if a set is compact.›

lemma homeomorphism_compact:
  fixes f :: "'a::topological_space ⇒ 'b::t2_space"
  assumes "compact S" "continuous_on S f"  "f ` S = T"  "inj_on f S"
  shows "∃g. homeomorphism S T f g"
proof -
  obtain g where g: "∀x∈S. g (f x) = x" "∀x∈T. f (g x) = x" "g ` T = S"
    using assms the_inv_into_f_f by fastforce 
  with assms show ?thesis
    unfolding homeomorphism_def homeomorphic_def by (metis continuous_on_inv)
qed

lemma homeomorphic_compact:
  fixes f :: "'a::topological_space ⇒ 'b::t2_space"
  shows "compact S ⟹ continuous_on S f ⟹ (f ` S = T) ⟹ inj_on f S ⟹ S homeomorphic T"
  unfolding homeomorphic_def by (metis homeomorphism_compact)

text‹Preservation of topological properties.›

lemma homeomorphic_compactness: "S homeomorphic T ⟹ (compact S ⟷ compact T)"
  unfolding homeomorphic_def homeomorphism_def
  by (metis compact_continuous_image)


subsection✐‹tag unimportant› ‹On Linorder Topologies›

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 ∈ T"
  then obtain c where c: "a < c" "{a..<c} ⊆ T"
    by (meson assms open_right)
  with assms dense[of a "min c b"]
  show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ 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 ∈ T"
  from open_left[OF this ‹a < b›]
  obtain c where c: "c < b" "{c<..b} ⊆ T" by auto
  with assms dense[of "max a c" b]
  show "∃y∈{a<..<b}. y ∈ T ∧ y ≠ 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 ⟹ closure {a <..< b} = {a .. b}" (is "_ ⟹ ?l = ?r")
proof
  have "?l ⊆ closure ?r"
    by (rule closure_mono) auto
  thus "closure {a<..<b} ⊆ {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} ∪ {b..}" by auto
  also have "closure … = {a..}" using ‹a < b› 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} ∪ {..a}" by auto
  also have "closure … = {..b}" using ‹a < b› 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} ∪ {a <..< b}" by auto
  also have "closure … = {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} ∪ {a <..< b}" by auto
  also have "closure … = {a .. b}" unfolding closure_Un
    by (auto simp: assms less_imp_le)
  finally show ?thesis .
qed

end