# Theory T1_Spaces

```section‹T1 and Hausdorff spaces›

theory T1_Spaces
imports Product_Topology
begin

section‹T1 spaces with equivalences to many naturally "nice" properties. ›

definition t1_space where
"t1_space X ≡ ∀x ∈ topspace X. ∀y ∈ topspace X. x≠y ⟶ (∃U. openin X U ∧ x ∈ U ∧ y ∉ U)"

lemma t1_space_expansive:
"⟦topspace Y = topspace X; ⋀U. openin X U ⟹ openin Y U⟧ ⟹ t1_space X ⟹ t1_space Y"
by (metis t1_space_def)

lemma t1_space_alt:
"t1_space X ⟷ (∀x ∈ topspace X. ∀y ∈ topspace X. x≠y ⟶ (∃U. closedin X U ∧ x ∈ U ∧ y ∉ U))"
by (metis DiffE DiffI closedin_def openin_closedin_eq t1_space_def)

lemma t1_space_empty [iff]: "t1_space trivial_topology"
by (simp add: t1_space_def)

lemma t1_space_derived_set_of_singleton:
"t1_space X ⟷ (∀x ∈ topspace X. X derived_set_of {x} = {})"
apply (simp add: t1_space_def derived_set_of_def, safe)
apply (metis openin_topspace)
by force

lemma t1_space_derived_set_of_finite:
"t1_space X ⟷ (∀S. finite S ⟶ X derived_set_of S = {})"
proof (intro iffI allI impI)
fix S :: "'a set"
assume "finite S"
then have fin: "finite ((λx. {x}) ` (topspace X ∩ S))"
by blast
assume "t1_space X"
then have "X derived_set_of (⋃x ∈ topspace X ∩ S. {x}) = {}"
unfolding derived_set_of_Union [OF fin]
by (auto simp: t1_space_derived_set_of_singleton)
then have "X derived_set_of (topspace X ∩ S) = {}"
by simp
then show "X derived_set_of S = {}"
by simp
qed (auto simp: t1_space_derived_set_of_singleton)

lemma t1_space_closedin_singleton:
"t1_space X ⟷ (∀x ∈ topspace X. closedin X {x})"
apply (rule iffI)
apply (simp add: closedin_contains_derived_set t1_space_derived_set_of_singleton)
using t1_space_alt by auto

lemma continuous_closed_imp_proper_map:
"⟦compact_space X; t1_space Y; continuous_map X Y f; closed_map X Y f⟧ ⟹ proper_map X Y f"
unfolding proper_map_def
by (smt (verit) closedin_compact_space closedin_continuous_map_preimage
Collect_cong singleton_iff t1_space_closedin_singleton)

lemma t1_space_euclidean: "t1_space (euclidean :: 'a::metric_space topology)"
by (simp add: t1_space_closedin_singleton)

lemma closedin_t1_singleton:
"⟦t1_space X; a ∈ topspace X⟧ ⟹ closedin X {a}"
by (simp add: t1_space_closedin_singleton)

lemma t1_space_closedin_finite:
"t1_space X ⟷ (∀S. finite S ∧ S ⊆ topspace X ⟶ closedin X S)"
apply (rule iffI)
apply (simp add: closedin_contains_derived_set t1_space_derived_set_of_finite)
by (simp add: t1_space_closedin_singleton)

lemma closure_of_singleton:
"t1_space X ⟹ X closure_of {a} = (if a ∈ topspace X then {a} else {})"
by (simp add: closure_of_eq t1_space_closedin_singleton closure_of_eq_empty_gen)

lemma separated_in_singleton:
assumes "t1_space X"
shows "separatedin X {a} S ⟷ a ∈ topspace X ∧ S ⊆ topspace X ∧ (a ∉ X closure_of S)"
"separatedin X S {a} ⟷ a ∈ topspace X ∧ S ⊆ topspace X ∧ (a ∉ X closure_of S)"
unfolding separatedin_def
using assms closure_of closure_of_singleton by fastforce+

lemma t1_space_openin_delete:
"t1_space X ⟷ (∀U x. openin X U ∧ x ∈ U ⟶ openin X (U - {x}))"
apply (rule iffI)
apply (meson closedin_t1_singleton in_mono openin_diff openin_subset)
by (simp add: closedin_def t1_space_closedin_singleton)

lemma t1_space_openin_delete_alt:
"t1_space X ⟷ (∀U x. openin X U ⟶ openin X (U - {x}))"
by (metis Diff_empty Diff_insert0 t1_space_openin_delete)

lemma t1_space_singleton_Inter_open:
"t1_space X ⟷ (∀x ∈ topspace X. ⋂{U. openin X U ∧ x ∈ U} = {x})"  (is "?P=?Q")
and t1_space_Inter_open_supersets:
"t1_space X ⟷ (∀S. S ⊆ topspace X ⟶ ⋂{U. openin X U ∧ S ⊆ U} = S)" (is "?P=?R")
proof -
have "?R ⟹ ?Q"
apply clarify
apply (drule_tac x="{x}" in spec, simp)
done
moreover have "?Q ⟹ ?P"
apply (clarsimp simp add: t1_space_def)
apply (drule_tac x=x in bspec)
apply (simp_all add: set_eq_iff)
by (metis (no_types, lifting))
moreover have "?P ⟹ ?R"
proof (clarsimp simp add: t1_space_closedin_singleton, rule subset_antisym)
fix S
assume S: "∀x∈topspace X. closedin X {x}" "S ⊆ topspace X"
then show "⋂ {U. openin X U ∧ S ⊆ U} ⊆ S"
apply clarsimp
by (metis Diff_insert_absorb Set.set_insert closedin_def openin_topspace subset_insert)
qed force
ultimately show "?P=?Q" "?P=?R"
by auto
qed

lemma t1_space_derived_set_of_infinite_openin:
"t1_space X ⟷
(∀S. X derived_set_of S =
{x ∈ topspace X. ∀U. x ∈ U ∧ openin X U ⟶ infinite(S ∩ U)})"
(is "_ = ?rhs")
proof
assume "t1_space X"
show ?rhs
proof safe
fix S x U
assume "x ∈ X derived_set_of S" "x ∈ U" "openin X U" "finite (S ∩ U)"
with ‹t1_space X› show "False"
apply (simp add: t1_space_derived_set_of_finite)
by (metis IntI empty_iff empty_subsetI inf_commute openin_Int_derived_set_of_subset subset_antisym)
next
fix S x
have eq: "(∃y. (y ≠ x) ∧ y ∈ S ∧ y ∈ T) ⟷ ~((S ∩ T) ⊆ {x})" for x S T
by blast
assume "x ∈ topspace X" "∀U. x ∈ U ∧ openin X U ⟶ infinite (S ∩ U)"
then show "x ∈ X derived_set_of S"
apply (clarsimp simp add: derived_set_of_def eq)
by (meson finite.emptyI finite.insertI finite_subset)
qed (auto simp: in_derived_set_of)
qed (auto simp: t1_space_derived_set_of_singleton)

lemma finite_t1_space_imp_discrete_topology:
"⟦topspace X = U; finite U; t1_space X⟧ ⟹ X = discrete_topology U"
by (metis discrete_topology_unique_derived_set t1_space_derived_set_of_finite)

lemma t1_space_subtopology: "t1_space X ⟹ t1_space(subtopology X U)"
by (simp add: derived_set_of_subtopology t1_space_derived_set_of_finite)

lemma closedin_derived_set_of_gen:
"t1_space X ⟹ closedin X (X derived_set_of S)"
apply (clarsimp simp add: in_derived_set_of closedin_contains_derived_set derived_set_of_subset_topspace)
by (metis DiffD2 insert_Diff insert_iff t1_space_openin_delete)

lemma derived_set_of_derived_set_subset_gen:
"t1_space X ⟹ X derived_set_of (X derived_set_of S) ⊆ X derived_set_of S"
by (meson closedin_contains_derived_set closedin_derived_set_of_gen)

lemma subtopology_eq_discrete_topology_gen_finite:
"⟦t1_space X; finite S⟧ ⟹ subtopology X S = discrete_topology(topspace X ∩ S)"
by (simp add: subtopology_eq_discrete_topology_gen t1_space_derived_set_of_finite)

lemma subtopology_eq_discrete_topology_finite:
"⟦t1_space X; S ⊆ topspace X; finite S⟧
⟹ subtopology X S = discrete_topology S"
by (simp add: subtopology_eq_discrete_topology_eq t1_space_derived_set_of_finite)

lemma t1_space_closed_map_image:
"⟦closed_map X Y f; f ` (topspace X) = topspace Y; t1_space X⟧ ⟹ t1_space Y"
by (metis closed_map_def finite_subset_image t1_space_closedin_finite)

lemma homeomorphic_t1_space: "X homeomorphic_space Y ⟹ (t1_space X ⟷ t1_space Y)"
apply (clarsimp simp add: homeomorphic_space_def)
by (meson homeomorphic_eq_everything_map homeomorphic_maps_map t1_space_closed_map_image)

proposition t1_space_product_topology:
"t1_space (product_topology X I)
⟷ (product_topology X I) = trivial_topology ∨ (∀i ∈ I. t1_space (X i))"
proof (cases "(product_topology X I) = trivial_topology")
case True
then show ?thesis
using True t1_space_empty by force
next
case False
then obtain f where f: "f ∈ (Π⇩E i∈I. topspace(X i))"
using discrete_topology_unique by (fastforce iff: null_topspace_iff_trivial)
have "t1_space (product_topology X I) ⟷ (∀i∈I. t1_space (X i))"
proof (intro iffI ballI)
show "t1_space (X i)" if "t1_space (product_topology X I)" and "i ∈ I" for i
proof -
have clo: "⋀h. h ∈ (Π⇩E i∈I. topspace (X i)) ⟹ closedin (product_topology X I) {h}"
using that by (simp add: t1_space_closedin_singleton)
show ?thesis
unfolding t1_space_closedin_singleton
proof clarify
show "closedin (X i) {xi}" if "xi ∈ topspace (X i)" for xi
using clo [of "λj ∈ I. if i=j then xi else f j"] f that ‹i ∈ I›
by (fastforce simp add: closedin_product_topology_singleton)
qed
qed
next
next
show "t1_space (product_topology X I)" if "∀i∈I. t1_space (X i)"
using that
by (simp add: t1_space_closedin_singleton Ball_def PiE_iff closedin_product_topology_singleton)
qed
then show ?thesis
using False by blast
qed

lemma t1_space_prod_topology:
"t1_space(prod_topology X Y) ⟷ (prod_topology X Y) = trivial_topology ∨ t1_space X ∧ t1_space Y"
proof (cases "(prod_topology X Y) = trivial_topology")
case True then show ?thesis
by auto
next
case False
have eq: "{(x,y)} = {x} × {y}" for x::'a and y::'b
by simp
have "t1_space (prod_topology X Y) ⟷ (t1_space X ∧ t1_space Y)"
using False
apply(simp add: t1_space_closedin_singleton closedin_prod_Times_iff eq
del: insert_Times_insert flip: null_topspace_iff_trivial ex_in_conv)
by blast
with False show ?thesis
by simp
qed

subsection‹Hausdorff Spaces›

definition Hausdorff_space
where
"Hausdorff_space X ≡
∀x y. x ∈ topspace X ∧ y ∈ topspace X ∧ (x ≠ y)
⟶ (∃U V. openin X U ∧ openin X V ∧ x ∈ U ∧ y ∈ V ∧ disjnt U V)"

lemma Hausdorff_space_expansive:
"⟦Hausdorff_space X; topspace X = topspace Y; ⋀U. openin X U ⟹ openin Y U⟧ ⟹ Hausdorff_space Y"
by (metis Hausdorff_space_def)

lemma Hausdorff_space_topspace_empty [iff]: "Hausdorff_space trivial_topology"
by (simp add: Hausdorff_space_def)

lemma Hausdorff_imp_t1_space:
"Hausdorff_space X ⟹ t1_space X"
by (metis Hausdorff_space_def disjnt_iff t1_space_def)

lemma closedin_derived_set_of:
"Hausdorff_space X ⟹ closedin X (X derived_set_of S)"
by (simp add: Hausdorff_imp_t1_space closedin_derived_set_of_gen)

lemma t1_or_Hausdorff_space:
"t1_space X ∨ Hausdorff_space X ⟷ t1_space X"
using Hausdorff_imp_t1_space by blast

lemma Hausdorff_space_sing_Inter_opens:
"⟦Hausdorff_space X; a ∈ topspace X⟧ ⟹ ⋂{u. openin X u ∧ a ∈ u} = {a}"
using Hausdorff_imp_t1_space t1_space_singleton_Inter_open by force

lemma Hausdorff_space_subtopology:
assumes "Hausdorff_space X" shows "Hausdorff_space(subtopology X S)"
proof -
have *: "disjnt U V ⟹ disjnt (S ∩ U) (S ∩ V)" for U V
by (simp add: disjnt_iff)
from assms show ?thesis
apply (simp add: Hausdorff_space_def openin_subtopology_alt)
apply (fast intro: * elim!: all_forward)
done
qed

lemma Hausdorff_space_compact_separation:
assumes X: "Hausdorff_space X" and S: "compactin X S" and T: "compactin X T" and "disjnt S T"
obtains U V where "openin X U" "openin X V" "S ⊆ U" "T ⊆ V" "disjnt U V"
proof (cases "S = {}")
case True
then show thesis
by (metis ‹compactin X T› compactin_subset_topspace disjnt_empty1 empty_subsetI openin_empty openin_topspace that)
next
case False
have "∀x ∈ S. ∃U V. openin X U ∧ openin X V ∧ x ∈ U ∧ T ⊆ V ∧ disjnt U V"
proof
fix a
assume "a ∈ S"
then have "a ∉ T"
by (meson assms(4) disjnt_iff)
have a: "a ∈ topspace X"
using S ‹a ∈ S› compactin_subset_topspace by blast
show "∃U V. openin X U ∧ openin X V ∧ a ∈ U ∧ T ⊆ V ∧ disjnt U V"
proof (cases "T = {}")
case True
then show ?thesis
using a disjnt_empty2 openin_empty by blast
next
case False
have "∀x ∈ topspace X - {a}. ∃U V. openin X U ∧ openin X V ∧ x ∈ U ∧ a ∈ V ∧ disjnt U V"
using X a by (simp add: Hausdorff_space_def)
then obtain U V where UV: "∀x ∈ topspace X - {a}. openin X (U x) ∧ openin X (V x) ∧ x ∈ U x ∧ a ∈ V x ∧ disjnt (U x) (V x)"
by metis
with ‹a ∉ T› compactin_subset_topspace [OF T]
have Topen: "∀W ∈ U ` T. openin X W" and Tsub: "T ⊆ ⋃ (U ` T)"
by auto
then obtain ℱ where ℱ: "finite ℱ" "ℱ ⊆ U ` T" and "T ⊆ ⋃ℱ"
using T unfolding compactin_def by meson
then obtain F where F: "finite F" "F ⊆ T" "ℱ = U ` F" and SUF: "T ⊆ ⋃(U ` F)" and "a ∉ F"
using finite_subset_image [OF ℱ] ‹a ∉ T› by (metis subsetD)
have U: "⋀x. ⟦x ∈ topspace X; x ≠ a⟧ ⟹ openin X (U x)"
and V: "⋀x. ⟦x ∈ topspace X; x ≠ a⟧ ⟹ openin X (V x)"
and disj: "⋀x. ⟦x ∈ topspace X; x ≠ a⟧ ⟹ disjnt (U x) (V x)"
using UV by blast+
show ?thesis
proof (intro exI conjI)
have "F ≠ {}"
using False SUF by blast
with ‹a ∉ F› show "openin X (⋂(V ` F))"
using F compactin_subset_topspace [OF T] by (force intro: V)
show "openin X (⋃(U ` F))"
using F Topen Tsub by (force intro: U)
show "disjnt (⋂(V ` F)) (⋃(U ` F))"
using disj
apply (auto simp: disjnt_def)
using ‹F ⊆ T› ‹a ∉ F› compactin_subset_topspace [OF T] by blast
show "a ∈ (⋂(V ` F))"
using ‹F ⊆ T› T UV ‹a ∉ T› compactin_subset_topspace by blast
qed (auto simp: SUF)
qed
qed
then obtain U V where UV: "∀x ∈ S. openin X (U x) ∧ openin X (V x) ∧ x ∈ U x ∧ T ⊆ V x ∧ disjnt (U x) (V x)"
by metis
then have "S ⊆ ⋃ (U ` S)"
by auto
moreover have "∀W ∈ U ` S. openin X W"
using UV by blast
ultimately obtain I where I: "S ⊆ ⋃ (U ` I)" "I ⊆ S" "finite I"
by (metis S compactin_def finite_subset_image)
show thesis
proof
show "openin X (⋃(U ` I))"
using ‹I ⊆ S› UV by blast
show "openin X (⋂ (V ` I))"
using False UV ‹I ⊆ S› ‹S ⊆ ⋃ (U ` I)› ‹finite I› by blast
show "disjnt (⋃(U ` I)) (⋂ (V ` I))"
by simp (meson UV ‹I ⊆ S› disjnt_subset2 in_mono le_INF_iff order_refl)
qed (use UV I in auto)
qed

lemma Hausdorff_space_compact_sets:
"Hausdorff_space X ⟷
(∀S T. compactin X S ∧ compactin X T ∧ disjnt S T
⟶ (∃U V. openin X U ∧ openin X V ∧ S ⊆ U ∧ T ⊆ V ∧ disjnt U V))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (meson Hausdorff_space_compact_separation)
next
assume R [rule_format]: ?rhs
show ?lhs
proof (clarsimp simp add: Hausdorff_space_def)
fix x y
assume "x ∈ topspace X" "y ∈ topspace X" "x ≠ y"
then show "∃U. openin X U ∧ (∃V. openin X V ∧ x ∈ U ∧ y ∈ V ∧ disjnt U V)"
using R [of "{x}" "{y}"] by auto
qed
qed

lemma compactin_imp_closedin:
assumes X: "Hausdorff_space X" and S: "compactin X S" shows "closedin X S"
proof -
have "S ⊆ topspace X"
by (simp add: assms compactin_subset_topspace)
moreover
have "∃T. openin X T ∧ x ∈ T ∧ T ⊆ topspace X - S" if "x ∈ topspace X" "x ∉ S" for x
using Hausdorff_space_compact_separation [OF X _ S, of "{x}"] that
apply (simp add: disjnt_def)
by (metis Diff_mono Diff_triv openin_subset)
ultimately show ?thesis
using closedin_def openin_subopen by force
qed

lemma closedin_Hausdorff_singleton:
"⟦Hausdorff_space X; x ∈ topspace X⟧ ⟹ closedin X {x}"
by (simp add: Hausdorff_imp_t1_space closedin_t1_singleton)

lemma closedin_Hausdorff_sing_eq:
"Hausdorff_space X ⟹ closedin X {x} ⟷ x ∈ topspace X"
by (meson closedin_Hausdorff_singleton closedin_subset insert_subset)

lemma Hausdorff_space_discrete_topology [simp]:
"Hausdorff_space (discrete_topology U)"
unfolding Hausdorff_space_def
by (metis Hausdorff_space_compact_sets Hausdorff_space_def compactin_discrete_topology equalityE openin_discrete_topology)

lemma compactin_Int:
"⟦Hausdorff_space X; compactin X S; compactin X T⟧ ⟹ compactin X (S ∩ T)"
by (simp add: closed_Int_compactin compactin_imp_closedin)

lemma finite_topspace_imp_discrete_topology:
"⟦topspace X = U; finite U; Hausdorff_space X⟧ ⟹ X = discrete_topology U"
using Hausdorff_imp_t1_space finite_t1_space_imp_discrete_topology by blast

lemma derived_set_of_finite:
"⟦Hausdorff_space X; finite S⟧ ⟹ X derived_set_of S = {}"
using Hausdorff_imp_t1_space t1_space_derived_set_of_finite by auto

lemma infinite_perfect_set:
"⟦Hausdorff_space X; S ⊆ X derived_set_of S; S ≠ {}⟧ ⟹ infinite S"
using derived_set_of_finite by blast

lemma derived_set_of_singleton:
"Hausdorff_space X ⟹ X derived_set_of {x} = {}"
by (simp add: derived_set_of_finite)

lemma closedin_Hausdorff_finite:
"⟦Hausdorff_space X; S ⊆ topspace X; finite S⟧ ⟹ closedin X S"
by (simp add: compactin_imp_closedin finite_imp_compactin_eq)

lemma open_in_Hausdorff_delete:
"⟦Hausdorff_space X; openin X S⟧ ⟹ openin X (S - {x})"
using Hausdorff_imp_t1_space t1_space_openin_delete_alt by auto

lemma closedin_Hausdorff_finite_eq:
"⟦Hausdorff_space X; finite S⟧ ⟹ closedin X S ⟷ S ⊆ topspace X"
by (meson closedin_Hausdorff_finite closedin_def)

lemma derived_set_of_infinite_openin:
"Hausdorff_space X
⟹ X derived_set_of S =
{x ∈ topspace X. ∀U. x ∈ U ∧ openin X U ⟶ infinite(S ∩ U)}"
using Hausdorff_imp_t1_space t1_space_derived_set_of_infinite_openin by fastforce

lemma Hausdorff_space_discrete_compactin:
"Hausdorff_space X
⟹ S ∩ X derived_set_of S = {} ∧ compactin X S ⟷ S ⊆ topspace X ∧ finite S"
using derived_set_of_finite discrete_compactin_eq_finite by fastforce

lemma Hausdorff_space_finite_topspace:
"Hausdorff_space X ⟹ X derived_set_of (topspace X) = {} ∧ compact_space X ⟷ finite(topspace X)"
using derived_set_of_finite discrete_compact_space_eq_finite by auto

lemma derived_set_of_derived_set_subset:
"Hausdorff_space X ⟹ X derived_set_of (X derived_set_of S) ⊆ X derived_set_of S"
by (simp add: Hausdorff_imp_t1_space derived_set_of_derived_set_subset_gen)

lemma Hausdorff_space_injective_preimage:
assumes "Hausdorff_space Y" and cmf: "continuous_map X Y f" and "inj_on f (topspace X)"
shows "Hausdorff_space X"
unfolding Hausdorff_space_def
proof clarify
fix x y
assume x: "x ∈ topspace X" and y: "y ∈ topspace X" and "x ≠ y"
then obtain U V where "openin Y U" "openin Y V" "f x ∈ U" "f y ∈ V" "disjnt U V"
using assms
by (smt (verit, ccfv_threshold) Hausdorff_space_def continuous_map image_subset_iff inj_on_def)
show "∃U V. openin X U ∧ openin X V ∧ x ∈ U ∧ y ∈ V ∧ disjnt U V"
proof (intro exI conjI)
show "openin X {x ∈ topspace X. f x ∈ U}"
using ‹openin Y U› cmf continuous_map by fastforce
show "openin X {x ∈ topspace X. f x ∈ V}"
using ‹openin Y V› cmf openin_continuous_map_preimage by blast
show "disjnt {x ∈ topspace X. f x ∈ U} {x ∈ topspace X. f x ∈ V}"
using ‹disjnt U V› by (auto simp add: disjnt_def)
qed (use x ‹f x ∈ U› y ‹f y ∈ V› in auto)
qed

lemma homeomorphic_Hausdorff_space:
"X homeomorphic_space Y ⟹ Hausdorff_space X ⟷ Hausdorff_space Y"
unfolding homeomorphic_space_def homeomorphic_maps_map
by (auto simp: homeomorphic_eq_everything_map Hausdorff_space_injective_preimage)

lemma Hausdorff_space_retraction_map_image:
"⟦retraction_map X Y r; Hausdorff_space X⟧ ⟹ Hausdorff_space Y"
unfolding retraction_map_def
using Hausdorff_space_subtopology homeomorphic_Hausdorff_space retraction_maps_section_image2 by blast

lemma compact_Hausdorff_space_optimal:
assumes eq: "topspace Y = topspace X" and XY: "⋀U. openin X U ⟹ openin Y U"
and "Hausdorff_space X" "compact_space Y"
shows "Y = X"
proof -
have "⋀U. closedin X U ⟹ closedin Y U"
using XY using topology_finer_closedin [OF eq]
by metis
have "openin Y S = openin X S" for S
by (metis XY assms(3) assms(4) closedin_compact_space compactin_contractive compactin_imp_closedin eq openin_closedin_eq)
then show ?thesis
by (simp add: topology_eq)
qed

lemma continuous_map_imp_closed_graph:
assumes f: "continuous_map X Y f" and Y: "Hausdorff_space Y"
shows "closedin (prod_topology X Y) ((λx. (x,f x)) ` topspace X)"
unfolding closedin_def
proof
show "(λx. (x, f x)) ` topspace X ⊆ topspace (prod_topology X Y)"
using continuous_map_def f by fastforce
show "openin (prod_topology X Y) (topspace (prod_topology X Y) - (λx. (x, f x)) ` topspace X)"
unfolding openin_prod_topology_alt
proof (intro allI impI)
show "∃U V. openin X U ∧ openin Y V ∧ x ∈ U ∧ y ∈ V ∧ U × V ⊆ topspace (prod_topology X Y) - (λx. (x, f x)) ` topspace X"
if "(x,y) ∈ topspace (prod_topology X Y) - (λx. (x, f x)) ` topspace X"
for x y
proof -
have "x ∈ topspace X" and y: "y ∈ topspace Y" "y ≠ f x"
using that by auto
then have "f x ∈ topspace Y"
using continuous_map_image_subset_topspace f by blast
then obtain U V where UV: "openin Y U" "openin Y V" "f x ∈ U" "y ∈ V" "disjnt U V"
using Y y Hausdorff_space_def by metis
show ?thesis
proof (intro exI conjI)
show "openin X {x ∈ topspace X. f x ∈ U}"
using ‹openin Y U› f openin_continuous_map_preimage by blast
show "{x ∈ topspace X. f x ∈ U} × V ⊆ topspace (prod_topology X Y) - (λx. (x, f x)) ` topspace X"
using UV by (auto simp: disjnt_iff dest: openin_subset)
qed (use UV ‹x ∈ topspace X› in auto)
qed
qed
qed

lemma continuous_imp_closed_map:
"⟦continuous_map X Y f; compact_space X; Hausdorff_space Y⟧ ⟹ closed_map X Y f"
by (meson closed_map_def closedin_compact_space compactin_imp_closedin image_compactin)

lemma continuous_imp_quotient_map:
"⟦continuous_map X Y f; compact_space X; Hausdorff_space Y; f ` (topspace X) = topspace Y⟧
⟹ quotient_map X Y f"
by (simp add: continuous_imp_closed_map continuous_closed_imp_quotient_map)

lemma continuous_imp_homeomorphic_map:
"⟦continuous_map X Y f; compact_space X; Hausdorff_space Y;
f ` (topspace X) = topspace Y; inj_on f (topspace X)⟧
⟹ homeomorphic_map X Y f"
by (simp add: continuous_imp_closed_map bijective_closed_imp_homeomorphic_map)

lemma continuous_imp_embedding_map:
"⟦continuous_map X Y f; compact_space X; Hausdorff_space Y; inj_on f (topspace X)⟧
⟹ embedding_map X Y f"
by (simp add: continuous_imp_closed_map injective_closed_imp_embedding_map)

lemma continuous_inverse_map:
assumes "compact_space X" "Hausdorff_space Y"
and cmf: "continuous_map X Y f" and gf: "⋀x. x ∈ topspace X ⟹ g(f x) = x"
and Sf:  "S ⊆ f ` (topspace X)"
shows "continuous_map (subtopology Y S) X g"
proof (rule continuous_map_from_subtopology_mono [OF _ ‹S ⊆ f ` (topspace X)›])
show "continuous_map (subtopology Y (f ` (topspace X))) X g"
unfolding continuous_map_closedin
proof (intro conjI ballI allI impI)
show "g ∈ topspace (subtopology Y (f ` topspace X)) → topspace X"
using gf by auto
next
fix C
assume C: "closedin X C"
show "closedin (subtopology Y (f ` topspace X))
{x ∈ topspace (subtopology Y (f ` topspace X)). g x ∈ C}"
proof (rule compactin_imp_closedin)
show "Hausdorff_space (subtopology Y (f ` topspace X))"
using Hausdorff_space_subtopology [OF ‹Hausdorff_space Y›] by blast
have "compactin Y (f ` C)"
using C cmf image_compactin closedin_compact_space [OF ‹compact_space X›] by blast
moreover have "{x ∈ topspace Y. x ∈ f ` topspace X ∧ g x ∈ C} = f ` C"
using closedin_subset [OF C] cmf by (auto simp: gf continuous_map_def)
ultimately have "compactin Y {x ∈ topspace Y. x ∈ f ` topspace X ∧ g x ∈ C}"
by simp
then show "compactin (subtopology Y (f ` topspace X))
{x ∈ topspace (subtopology Y (f ` topspace X)). g x ∈ C}"
by (auto simp add: compactin_subtopology)
qed
qed
qed

lemma closed_map_paired_continuous_map_right:
"⟦continuous_map X Y f; Hausdorff_space Y⟧ ⟹ closed_map X (prod_topology X Y) (λx. (x,f x))"
by (simp add: continuous_map_imp_closed_graph embedding_map_graph embedding_imp_closed_map)

lemma closed_map_paired_continuous_map_left:
assumes f: "continuous_map X Y f" and Y: "Hausdorff_space Y"
shows "closed_map X (prod_topology Y X) (λx. (f x,x))"
proof -
have eq: "(λx. (f x,x)) = (λ(a,b). (b,a)) ∘ (λx. (x,f x))"
by auto
show ?thesis
unfolding eq
proof (rule closed_map_compose)
show "closed_map X (prod_topology X Y) (λx. (x, f x))"
using Y closed_map_paired_continuous_map_right f by blast
show "closed_map (prod_topology X Y) (prod_topology Y X) (λ(a, b). (b, a))"
by (metis homeomorphic_map_swap homeomorphic_imp_closed_map)
qed
qed

lemma proper_map_paired_continuous_map_right:
"⟦continuous_map X Y f; Hausdorff_space Y⟧
⟹ proper_map X (prod_topology X Y) (λx. (x,f x))"
using closed_injective_imp_proper_map closed_map_paired_continuous_map_right
by (metis (mono_tags, lifting) Pair_inject inj_onI)

lemma proper_map_paired_continuous_map_left:
"⟦continuous_map X Y f; Hausdorff_space Y⟧
⟹ proper_map X (prod_topology Y X) (λx. (f x,x))"
using closed_injective_imp_proper_map closed_map_paired_continuous_map_left
by (metis (mono_tags, lifting) Pair_inject inj_onI)

lemma Hausdorff_space_prod_topology:
"Hausdorff_space(prod_topology X Y) ⟷ (prod_topology X Y) = trivial_topology ∨ Hausdorff_space X ∧ Hausdorff_space Y"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (rule topological_property_of_prod_component) (auto simp: Hausdorff_space_subtopology homeomorphic_Hausdorff_space)
next
assume R: ?rhs
show ?lhs
proof (cases "(topspace X × topspace Y) = {}")
case False
with R have ne: "topspace X ≠ {}" "topspace Y ≠ {}" and X: "Hausdorff_space X" and Y: "Hausdorff_space Y"
by auto
show ?thesis
unfolding Hausdorff_space_def
proof clarify
fix x y x' y'
assume xy: "(x, y) ∈ topspace (prod_topology X Y)"
and xy': "(x',y') ∈ topspace (prod_topology X Y)"
and *: "∄U V. openin (prod_topology X Y) U ∧ openin (prod_topology X Y) V
∧ (x, y) ∈ U ∧ (x', y') ∈ V ∧ disjnt U V"
have False if "x ≠ x' ∨ y ≠ y'"
using that
proof
assume "x ≠ x'"
then obtain U V where "openin X U" "openin X V" "x ∈ U" "x' ∈ V" "disjnt U V"
by (metis Hausdorff_space_def X mem_Sigma_iff topspace_prod_topology xy xy')
let ?U = "U × topspace Y"
let ?V = "V × topspace Y"
have "openin (prod_topology X Y) ?U" "openin (prod_topology X Y) ?V"
by (simp_all add: openin_prod_Times_iff ‹openin X U› ‹openin X V›)
moreover have "disjnt ?U ?V"
by (simp add: ‹disjnt U V›)
ultimately show False
using * ‹x ∈ U› ‹x' ∈ V› xy xy' by (metis SigmaD2 SigmaI topspace_prod_topology)
next
assume "y ≠ y'"
then obtain U V where "openin Y U" "openin Y V" "y ∈ U" "y' ∈ V" "disjnt U V"
by (metis Hausdorff_space_def Y mem_Sigma_iff topspace_prod_topology xy xy')
let ?U = "topspace X × U"
let ?V = "topspace X × V"
have "openin (prod_topology X Y) ?U" "openin (prod_topology X Y) ?V"
by (simp_all add: openin_prod_Times_iff ‹openin Y U› ‹openin Y V›)
moreover have "disjnt ?U ?V"
by (simp add: ‹disjnt U V›)
ultimately show False
using "*" ‹y ∈ U› ‹y' ∈ V› xy xy' by (metis SigmaD1 SigmaI topspace_prod_topology)
qed
then show "x = x' ∧ y = y'"
by blast
qed
qed force
qed

lemma Hausdorff_space_product_topology:
"Hausdorff_space (product_topology X I) ⟷ (Π⇩E i∈I. topspace(X i)) = {} ∨ (∀i ∈ I. Hausdorff_space (X i))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
by (simp add: Hausdorff_space_subtopology PiE_eq_empty_iff homeomorphic_Hausdorff_space
topological_property_of_product_component)
next
assume R: ?rhs
show ?lhs
proof (cases "(Π⇩E i∈I. topspace(X i)) = {}")
case True
then show ?thesis
by (simp add: Hausdorff_space_def)
next
case False
have "∃U V. openin (product_topology X I) U ∧ openin (product_topology X I) V ∧ f ∈ U ∧ g ∈ V ∧ disjnt U V"
if f: "f ∈ (Π⇩E i∈I. topspace (X i))" and g: "g ∈ (Π⇩E i∈I. topspace (X i))" and "f ≠ g"
for f g :: "'a ⇒ 'b"
proof -
obtain m where "f m ≠ g m"
using ‹f ≠ g› by blast
then have "m ∈ I"
using f g by fastforce
then have "Hausdorff_space (X m)"
using False that R by blast
then obtain U V where U: "openin (X m) U" and V: "openin (X m) V" and "f m ∈ U" "g m ∈ V" "disjnt U V"
by (metis Hausdorff_space_def PiE_mem ‹f m ≠ g m› ‹m ∈ I› f g)
show ?thesis
proof (intro exI conjI)
let ?U = "(Π⇩E i∈I. topspace(X i)) ∩ {x. x m ∈ U}"
let ?V = "(Π⇩E i∈I. topspace(X i)) ∩ {x. x m ∈ V}"
show "openin (product_topology X I) ?U" "openin (product_topology X I) ?V"
using ‹m ∈ I› U V
by (force simp add: openin_product_topology intro: arbitrary_union_of_inc relative_to_inc finite_intersection_of_inc)+
show "f ∈ ?U"
using ‹f m ∈ U› f by blast
show "g ∈ ?V"
using ‹g m ∈ V› g by blast
show "disjnt ?U ?V"
using ‹disjnt U V› by (auto simp: PiE_def Pi_def disjnt_def)
qed
qed
then show ?thesis
by (simp add: Hausdorff_space_def)
qed
qed

lemma Hausdorff_space_closed_neighbourhood:
"Hausdorff_space X ⟷
(∀x ∈ topspace X. ∃U C. openin X U ∧ closedin X C ∧
Hausdorff_space(subtopology X C) ∧ x ∈ U ∧ U ⊆ C)" (is "_ = ?rhs")
proof
assume R: ?rhs
show "Hausdorff_space X"
unfolding Hausdorff_space_def
proof clarify
fix x y
assume x: "x ∈ topspace X" and y: "y ∈ topspace X" and "x ≠ y"
obtain T C where *: "openin X T" "closedin X C" "x ∈ T" "T ⊆ C"
and C: "Hausdorff_space (subtopology X C)"
by (meson R ‹x ∈ topspace X›)
show "∃U V. openin X U ∧ openin X V ∧ x ∈ U ∧ y ∈ V ∧ disjnt U V"
proof (cases "y ∈ C")
case True
with * C obtain U V where U: "openin (subtopology X C) U"
and V: "openin (subtopology X C) V"
and "x ∈ U" "y ∈ V" "disjnt U V"
unfolding Hausdorff_space_def
by (smt (verit, best) ‹x ≠ y› closedin_subset subsetD topspace_subtopology_subset)
then obtain U' V' where UV': "U = U' ∩ C" "openin X U'" "V = V' ∩ C" "openin X V'"
by (meson openin_subtopology)
have "disjnt (T ∩ U') V'"
using ‹disjnt U V› UV' ‹T ⊆ C› by (force simp: disjnt_iff)
with ‹T ⊆ C› have "disjnt (T ∩ U') (V' ∪ (topspace X - C))"
unfolding disjnt_def by blast
moreover
have "openin X (T ∩ U')"
by (simp add: ‹openin X T› ‹openin X U'› openin_Int)
moreover have "openin X (V' ∪ (topspace X - C))"
using ‹closedin X C› ‹openin X V'› by auto
ultimately show ?thesis
using UV' ‹x ∈ T› ‹x ∈ U› ‹y ∈ V› by blast
next
case False
with * y show ?thesis
by (force simp: closedin_def disjnt_def)
qed
qed
qed fastforce

end
```