# Theory 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"

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

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

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

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

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

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"

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

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})"
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

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)
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"
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"

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"

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..})"
then have "l' islimpt (f ` {N..})"
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)"

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)"

lemma frequently_at_imp_islimpt:
assumes "frequently (λy. y ∈ A) (at x)"
shows   "x islimpt A"

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)"

lemma closed_closure[simp, intro]: "closed (closure S)"

lemma closure_subset: "S ⊆ closure S"

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"

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"

lemma interior_complement: "interior (- S) = - closure S"

lemma interior_diff: "interior(S - T) = interior S - closure T"

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 (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)"

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"
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 {} = {}"

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]

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```