Theory HOL-Analysis.Abstract_Topology_2

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

section ‹Abstract Topology 2›

theory Abstract_Topology_2
  imports
    Elementary_Topology Abstract_Topology Continuum_Not_Denumerable
    "HOL-Library.Indicator_Function"
    "HOL-Library.Equipollence"
begin

text ‹Combination of Elementary and Abstract Topology›

lemma approachable_lt_le2: 
  "(∃(d::real) > 0. ∀x. Q x ⟶ f x < d ⟶ P x) ⟷ (∃d>0. ∀x. f x ≤ d ⟶ Q x ⟶ P x)"
  by (meson dense less_eq_real_def order_le_less_trans)

lemma triangle_lemma:
  fixes x y z :: real
  assumes x: "0 ≤ x"
    and y: "0 ≤ y"
    and z: "0 ≤ z"
    and xy: "x⇧2 ≤ y⇧2 + z⇧2"
  shows "x ≤ y + z"
proof -
  have "y⇧2 + z⇧2 ≤ y⇧2 + 2 * y * z + z⇧2"
    using z y by simp
  with xy have th: "x⇧2 ≤ (y + z)⇧2"
    by (simp add: power2_eq_square field_simps)
  from y z have yz: "y + z ≥ 0"
    by arith
  from power2_le_imp_le[OF th yz] show ?thesis .
qed

lemma isCont_indicator:
  fixes x :: "'a::t2_space"
  shows "isCont (indicator A :: 'a ⇒ real) x = (x ∉ frontier A)"
proof auto
  fix x
  assume cts_at: "isCont (indicator A :: 'a ⇒ real) x" and fr: "x ∈ frontier A"
  with continuous_at_open have 1: "∀V::real set. open V ∧ indicator A x ∈ V ⟶
    (∃U::'a set. open U ∧ x ∈ U ∧ (∀y∈U. indicator A y ∈ V))" by auto
  show False
  proof (cases "x ∈ A")
    assume x: "x ∈ A"
    hence "indicator A x ∈ ({0<..<2} :: real set)" by simp
    with 1 obtain U where U: "open U" "x ∈ U" "∀y∈U. indicator A y ∈ ({0<..<2} :: real set)"
      using open_greaterThanLessThan by metis
    hence "∀y∈U. indicator A y > (0::real)"
      unfolding greaterThanLessThan_def by auto
    hence "U ⊆ A" using indicator_eq_0_iff by force
    hence "x ∈ interior A" using U interiorI by auto
    thus ?thesis using fr unfolding frontier_def by simp
  next
    assume x: "x ∉ A"
    hence "indicator A x ∈ ({-1<..<1} :: real set)" by simp
    with 1 obtain U where U: "open U" "x ∈ U" "∀y∈U. indicator A y ∈ ({-1<..<1} :: real set)"
      using 1 open_greaterThanLessThan by metis
    hence "∀y∈U. indicator A y < (1::real)"
      unfolding greaterThanLessThan_def by auto
    hence "U ⊆ -A" by auto
    hence "x ∈ interior (-A)" using U interiorI by auto
    thus ?thesis using fr interior_complement unfolding frontier_def by auto
  qed
next
  assume nfr: "x ∉ frontier A"
  hence "x ∈ interior A ∨ x ∈ interior (-A)"
    by (auto simp: frontier_def closure_interior)
  thus "isCont ((indicator A)::'a ⇒ real) x"
  proof
    assume int: "x ∈ interior A"
    then obtain U where U: "open U" "x ∈ U" "U ⊆ A" unfolding interior_def by auto
    hence "∀y∈U. indicator A y = (1::real)" unfolding indicator_def by auto
    hence "continuous_on U (indicator A)" by (simp add: indicator_eq_1_iff)
    thus ?thesis using U continuous_on_eq_continuous_at by auto
  next
    assume ext: "x ∈ interior (-A)"
    then obtain U where U: "open U" "x ∈ U" "U ⊆ -A" unfolding interior_def by auto
    then have "continuous_on U (indicator A)"
      using continuous_on_topological by (auto simp: subset_iff)
    thus ?thesis using U continuous_on_eq_continuous_at by auto
  qed
qed

lemma islimpt_closure:
  "⟦S ⊆ T; ⋀x. ⟦x islimpt S; x ∈ T⟧ ⟹ x ∈ S⟧ ⟹ S = T ∩ closure S"
  using closure_def by fastforce

lemma closedin_limpt:
  "closedin (top_of_set T) S ⟷ S ⊆ T ∧ (∀x. x islimpt S ∧ x ∈ T ⟶ x ∈ S)"
proof -
  have "⋀U x. ⟦closed U; S = T ∩ U; x islimpt S; x ∈ T⟧ ⟹ x ∈ S"
    by (meson IntI closed_limpt inf_le2 islimpt_subset)
  then show ?thesis
    by (metis closed_closure closedin_closed closedin_imp_subset islimpt_closure)
qed

lemma closedin_closed_eq: "closed S ⟹ closedin (top_of_set S) T ⟷ closed T ∧ T ⊆ S"
  by (meson closedin_limpt closed_subset closedin_closed_trans)

lemma connected_closed_set:
   "closed S
    ⟹ connected S ⟷ (∄A B. closed A ∧ closed B ∧ A ≠ {} ∧ B ≠ {} ∧ A ∪ B = S ∧ A ∩ B = {})"
  unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast

text ‹If a connnected set is written as the union of two nonempty closed sets, 
  then these sets have to intersect.›

lemma connected_as_closed_union:
  assumes "connected C" "C = A ∪ B" "closed A" "closed B" "A ≠ {}" "B ≠ {}"
  shows "A ∩ B ≠ {}"
  by (metis assms closed_Un connected_closed_set)

lemma closedin_subset_trans:
  "closedin (top_of_set U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹
    closedin (top_of_set T) S"
  by (meson closedin_limpt subset_iff)

lemma openin_subset_trans:
  "openin (top_of_set U) S ⟹ S ⊆ T ⟹ T ⊆ U ⟹
    openin (top_of_set T) S"
  by (auto simp: openin_open)

lemma closedin_compact:
  "⟦compact S; closedin (top_of_set S) T⟧ ⟹ compact T"
  by (metis closedin_closed compact_Int_closed)

lemma closedin_compact_eq:
  fixes S :: "'a::t2_space set"
  shows "compact S ⟹ (closedin (top_of_set S) T ⟷ compact T ∧ T ⊆ S)"
  by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)


subsection ‹Closure›

lemma euclidean_closure_of [simp]: "euclidean closure_of S = closure S"
  by (auto simp: closure_of_def closure_def islimpt_def)

lemma closure_openin_Int_closure:
  assumes ope: "openin (top_of_set U) S" and "T ⊆ U"
  shows "closure(S ∩ closure T) = closure(S ∩ T)"
proof
  obtain V where "open V" and S: "S = U ∩ V"
    using ope using openin_open by metis
  show "closure (S ∩ closure T) ⊆ closure (S ∩ T)"
    proof (clarsimp simp: S)
      fix x
      assume  "x ∈ closure (U ∩ V ∩ closure T)"
      then have "V ∩ closure T ⊆ A ⟹ x ∈ closure A" for A
          by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
      then have "x ∈ closure (T ∩ V)"
         by (metis ‹open V› closure_closure inf_commute open_Int_closure_subset)
      then show "x ∈ closure (U ∩ V ∩ T)"
        by (metis ‹T ⊆ U› inf.absorb_iff2 inf_assoc inf_commute)
    qed
next
  show "closure (S ∩ T) ⊆ closure (S ∩ closure T)"
    by (meson Int_mono closure_mono closure_subset order_refl)
qed

corollary infinite_openin:
  fixes S :: "'a :: t1_space set"
  shows "⟦openin (top_of_set U) S; x ∈ S; x islimpt U⟧ ⟹ infinite S"
  by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)

lemma closure_Int_ballI:
  assumes "⋀U. ⟦openin (top_of_set S) U; U ≠ {}⟧ ⟹ T ∩ U ≠ {}"
  shows "S ⊆ closure T"
proof (clarsimp simp: closure_iff_nhds_not_empty)
  fix x and A and V
  assume "x ∈ S" "V ⊆ A" "open V" "x ∈ V" "T ∩ A = {}"
  then have "openin (top_of_set S) (A ∩ V ∩ S)"
    by (simp add: inf_absorb2 openin_subtopology_Int)
  moreover have "A ∩ V ∩ S ≠ {}" using ‹x ∈ V› ‹V ⊆ A› ‹x ∈ S›
    by auto
  ultimately show False
    using ‹T ∩ A = {}› assms by fastforce
qed


subsection ‹Frontier›

lemma euclidean_interior_of [simp]: "euclidean interior_of S = interior S"
  by (auto simp: interior_of_def interior_def)

lemma euclidean_frontier_of [simp]: "euclidean frontier_of S = frontier S"
  by (auto simp: frontier_of_def frontier_def)

lemma connected_Int_frontier:
     "⟦connected S; S ∩ T ≠ {}; S - T ≠ {}⟧ ⟹ S ∩ frontier T ≠ {}"
  apply (simp add: frontier_interiors connected_openin, safe)
  apply (drule_tac x="S ∩ interior T" in spec, safe)
   apply (drule_tac [2] x="S ∩ interior (-T)" in spec)
   apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
  done

subsection ‹Compactness›

lemma openin_delete:
  fixes a :: "'a :: t1_space"
  shows "openin (top_of_set u) S ⟹ openin (top_of_set u) (S - {a})"
by (metis Int_Diff open_delete openin_open)

lemma compact_eq_openin_cover:
  "compact S ⟷
    (∀C. (∀c∈C. openin (top_of_set S) c) ∧ S ⊆ ⋃C ⟶
      (∃D⊆C. finite D ∧ S ⊆ ⋃D))"
proof safe
  fix C
  assume "compact S" and "∀c∈C. openin (top_of_set S) c" and "S ⊆ ⋃C"
  then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ ⋃{T. open T ∧ S ∩ T ∈ C}"
    unfolding openin_open by force+
  with ‹compact S› obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ ⋃D"
    by (meson compactE)
  then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ ⋃(image (λT. S ∩ T) D)"
    by auto
  then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
next
  assume 1: "∀C. (∀c∈C. openin (top_of_set S) c) ∧ S ⊆ ⋃C ⟶
        (∃D⊆C. finite D ∧ S ⊆ ⋃D)"
  show "compact S"
  proof (rule compactI)
    fix C
    let ?C = "image (λT. S ∩ T) C"
    assume "∀t∈C. open t" and "S ⊆ ⋃C"
    then have "(∀c∈?C. openin (top_of_set S) c) ∧ S ⊆ ⋃?C"
      unfolding openin_open by auto
    with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ ⋃D"
      by metis
    let ?D = "inv_into C (λT. S ∩ T) ` D"
    have "?D ⊆ C ∧ finite ?D ∧ S ⊆ ⋃?D"
    proof (intro conjI)
      from ‹D ⊆ ?C› show "?D ⊆ C"
        by (fast intro: inv_into_into)
      from ‹finite D› show "finite ?D"
        by (rule finite_imageI)
      from ‹S ⊆ ⋃D› show "S ⊆ ⋃?D"
        by (metis ‹D ⊆ (∩) S ` C› image_inv_into_cancel inf_Sup le_infE)
    qed
    then show "∃D⊆C. finite D ∧ S ⊆ ⋃D" ..
  qed
qed


subsection ‹Continuity›

lemma interior_image_subset:
  assumes "inj f" "⋀x. continuous (at x) f"
  shows "interior (f ` S) ⊆ f ` (interior S)"
proof
  fix x assume "x ∈ interior (f ` S)"
  then obtain T where as: "open T" "x ∈ T" "T ⊆ f ` S" ..
  then have "x ∈ f ` S" by auto
  then obtain y where y: "y ∈ S" "x = f y" by auto
  have "open (f -` T)"
    using assms ‹open T› by (simp add: continuous_at_imp_continuous_on open_vimage)
  moreover have "y ∈ vimage f T"
    using ‹x = f y› ‹x ∈ T› by simp
  moreover have "vimage f T ⊆ S"
    using ‹T ⊆ image f S› ‹inj f› unfolding inj_on_def subset_eq by auto
  ultimately have "y ∈ interior S" ..
  with ‹x = f y› show "x ∈ f ` interior S" ..
qed

subsection✐‹tag unimportant› ‹Equality of continuous functions on closure and related results›

lemma continuous_closedin_preimage_constant:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "continuous_on S f ⟹ closedin (top_of_set S) {x ∈ S. f x = a}"
  using continuous_closedin_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

lemma continuous_closed_preimage_constant:
  fixes f :: "_ ⇒ 'b::t1_space"
  shows "continuous_on S f ⟹ closed S ⟹ closed {x ∈ S. f x = a}"
  using continuous_closed_preimage[of S f "{a}"] by (simp add: vimage_def Collect_conj_eq)

lemma continuous_constant_on_closure:
  fixes f :: "_ ⇒ 'b::t1_space"
  assumes "continuous_on (closure S) f"
      and "⋀x. x ∈ S ⟹ f x = a"
      and "x ∈ closure S"
  shows "f x = a"
    using continuous_closed_preimage_constant[of "closure S" f a]
      assms closure_minimal[of S "{x ∈ closure S. f x = a}"] closure_subset
    unfolding subset_eq
    by auto

lemma image_closure_subset:
  assumes contf: "continuous_on (closure S) f"
    and "closed T"
    and "(f ` S) ⊆ T"
  shows "f ` (closure S) ⊆ T"
proof -
  have "S ⊆ {x ∈ closure S. f x ∈ T}"
    using assms(3) closure_subset by auto
  moreover have "closed (closure S ∩ f -` T)"
    using continuous_closed_preimage[OF contf] ‹closed T› by auto
  ultimately have "closure S = (closure S ∩ f -` T)"
    using closure_minimal[of S "(closure S ∩ f -` T)"] by auto
  then show ?thesis by auto
qed

lemma continuous_image_closure_subset:
  assumes "continuous_on A f" "closure B ⊆ A"
  shows   "f ` closure B ⊆ closure (f ` B)"
  using assms by (meson closed_closure closure_subset continuous_on_subset image_closure_subset)

subsection✐‹tag unimportant› ‹A function constant on a set›

definition constant_on  (infixl "(constant'_on)" 50)
  where "f constant_on A ≡ ∃y. ∀x∈A. f x = y"

lemma constant_on_subset: "⟦f constant_on A; B ⊆ A⟧ ⟹ f constant_on B"
  unfolding constant_on_def by blast

lemma injective_not_constant:
  fixes S :: "'a::{perfect_space} set"
  shows "⟦open S; inj_on f S; f constant_on S⟧ ⟹ S = {}"
  unfolding constant_on_def
  by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)

lemma constant_on_compose:
  assumes "f constant_on A"
  shows   "g ∘ f constant_on A"
  using assms by (auto simp: constant_on_def)

lemma not_constant_onI:
  "f x ≠ f y ⟹ x ∈ A ⟹ y ∈ A ⟹ ¬f constant_on A"
  unfolding constant_on_def by metis

lemma constant_onE:
  assumes "f constant_on S" and "⋀x. x∈S ⟹ f x = g x"
  shows "g constant_on S"
  using assms unfolding constant_on_def by simp

lemma constant_on_closureI:
  fixes f :: "_ ⇒ 'b::t1_space"
  assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
  shows "f constant_on (closure S)"
  using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
  by metis


subsection✐‹tag unimportant› ‹Continuity relative to a union.›

lemma continuous_on_Un_local:
    "⟦closedin (top_of_set (S ∪ T)) S; closedin (top_of_set (S ∪ T)) T;
      continuous_on S f; continuous_on T f⟧
     ⟹ continuous_on (S ∪ T) f"
  unfolding continuous_on closedin_limpt
  by (metis Lim_trivial_limit Lim_within_Un Un_iff trivial_limit_within)

lemma continuous_on_cases_local:
     "⟦closedin (top_of_set (S ∪ T)) S; closedin (top_of_set (S ∪ T)) T;
       continuous_on S f; continuous_on T g;
       ⋀x. ⟦x ∈ S ∧ ¬P x ∨ x ∈ T ∧ P x⟧ ⟹ f x = g x⟧
      ⟹ continuous_on (S ∪ T) (λx. if P x then f x else g x)"
  by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)

lemma continuous_on_cases_le:
  fixes h :: "'a :: topological_space ⇒ real"
  assumes "continuous_on {x ∈ S. h x ≤ a} f"
      and "continuous_on {x ∈ S. a ≤ h x} g"
      and h: "continuous_on S h"
      and "⋀x. ⟦x ∈ S; h x = a⟧ ⟹ f x = g x"
    shows "continuous_on S (λx. if h x ≤ a then f(x) else g(x))"
proof -
  have S: "S = (S ∩ h -` atMost a) ∪ (S ∩ h -` atLeast a)"
    by force
  have 1: "closedin (top_of_set S) (S ∩ h -` atMost a)"
    by (rule continuous_closedin_preimage [OF h closed_atMost])
  have 2: "closedin (top_of_set S) (S ∩ h -` atLeast a)"
    by (rule continuous_closedin_preimage [OF h closed_atLeast])
  have [simp]: "S ∩ h -` {..a} = {x ∈ S. h x ≤ a}" "S ∩ h -` {a..} = {x ∈ S. a ≤ h x}"
    by auto
  have "continuous_on (S ∩ h -` {..a} ∪ S ∩ h -` {a..}) (λx. if h x ≤ a then f x else g x)"
    by (intro continuous_on_cases_local) (use 1 2 S assms in auto)
  then show ?thesis
    using S by force
qed

lemma continuous_on_cases_1:
  fixes S :: "real set"
  assumes "continuous_on {t ∈ S. t ≤ a} f"
      and "continuous_on {t ∈ S. a ≤ t} g"
      and "a ∈ S ⟹ f a = g a"
    shows "continuous_on S (λt. if t ≤ a then f(t) else g(t))"
using assms
by (auto intro: continuous_on_cases_le [where h = id, simplified])


subsection✐‹tag unimportant›‹Inverse function property for open/closed maps›

lemma continuous_on_inverse_open_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "⋀x. x ∈ S ⟹ g (f x) = x"
    and oo: "⋀U. openin (top_of_set S) U ⟹ openin (top_of_set T) (f ` U)"
  shows "continuous_on T g"
proof -
  from imf injf have gTS: "g ` T = S"
    by force
  from imf injf have fU: "U ⊆ S ⟹ (f ` U) = T ∩ g -` U" for U
    by force
  show ?thesis
    by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
qed

lemma continuous_on_inverse_closed_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "⋀x. x ∈ S ⟹ g(f x) = x"
    and oo: "⋀U. closedin (top_of_set S) U ⟹ closedin (top_of_set T) (f ` U)"
  shows "continuous_on T g"
proof -
  from imf injf have gTS: "g ` T = S"
    by force
  from imf injf have fU: "U ⊆ S ⟹ (f ` U) = T ∩ g -` U" for U
    by force
  show ?thesis
    by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
qed

lemma homeomorphism_injective_open_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "inj_on f S"
    and oo: "⋀U. openin (top_of_set S) U ⟹ openin (top_of_set T) (f ` U)"
  obtains g where "homeomorphism S T f g"
proof
  have "continuous_on T (inv_into S f)"
    by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
  with imf injf contf show "homeomorphism S T f (inv_into S f)"
    by (auto simp: homeomorphism_def)
qed

lemma homeomorphism_injective_closed_map:
  assumes contf: "continuous_on S f"
    and imf: "f ` S = T"
    and injf: "inj_on f S"
    and oo: "⋀U. closedin (top_of_set S) U ⟹ closedin (top_of_set T) (f ` U)"
  obtains g where "homeomorphism S T f g"
proof
  have "continuous_on T (inv_into S f)"
    by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
  with imf injf contf show "homeomorphism S T f (inv_into S f)"
    by (auto simp: homeomorphism_def)
qed

lemma homeomorphism_imp_open_map:
  assumes hom: "homeomorphism S T f g"
    and oo: "openin (top_of_set S) U"
  shows "openin (top_of_set T) (f ` U)"
proof -
  from hom oo have [simp]: "f ` U = T ∩ g -` U"
    using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
  from hom have "continuous_on T g"
    unfolding homeomorphism_def by blast
  moreover have "g ` T = S"
    by (metis hom homeomorphism_def)
  ultimately show ?thesis
    by (simp add: continuous_on_open oo)
qed

lemma homeomorphism_imp_closed_map:
  assumes hom: "homeomorphism S T f g"
    and oo: "closedin (top_of_set S) U"
  shows "closedin (top_of_set T) (f ` U)"
proof -
  from hom oo have [simp]: "f ` U = T ∩ g -` U"
    using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
  from hom have "continuous_on T g"
    unfolding homeomorphism_def by blast
  moreover have "g ` T = S"
    by (metis hom homeomorphism_def)
  ultimately show ?thesis
    by (simp add: continuous_on_closed oo)
qed

subsection✐‹tag unimportant› ‹Seperability›

lemma subset_second_countable:
  obtains ℬ :: "'a:: second_countable_topology set set"
    where "countable ℬ"
          "{} ∉ ℬ"
          "⋀C. C ∈ ℬ ⟹ openin(top_of_set S) C"
          "⋀T. openin(top_of_set S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
proof -
  obtain ℬ :: "'a set set"
    where "countable ℬ"
      and opeB: "⋀C. C ∈ ℬ ⟹ openin(top_of_set S) C"
      and ℬ:    "⋀T. openin(top_of_set S) T ⟹ ∃𝒰. 𝒰 ⊆ ℬ ∧ T = ⋃𝒰"
  proof -
    obtain 𝒞 :: "'a set set"
      where "countable 𝒞" and ope: "⋀C. C ∈ 𝒞 ⟹ open C"
        and 𝒞: "⋀S. open S ⟹ ∃U. U ⊆ 𝒞 ∧ S = ⋃U"
      by (metis univ_second_countable that)
    show ?thesis
    proof
      show "countable ((λC. S ∩ C) ` 𝒞)"
        by (simp add: ‹countable 𝒞›)
      show "⋀C. C ∈ (∩) S ` 𝒞 ⟹ openin (top_of_set S) C"
        using ope by auto
      show "⋀T. openin (top_of_set S) T ⟹ ∃𝒰⊆(∩) S ` 𝒞. T = ⋃𝒰"
        by (metis 𝒞 image_mono inf_Sup openin_open)
    qed
  qed
  show ?thesis
  proof
    show "countable (ℬ - {{}})"
      using ‹countable ℬ› by blast
    show "⋀C. ⟦C ∈ ℬ - {{}}⟧ ⟹ openin (top_of_set S) C"
      by (simp add: ‹⋀C. C ∈ ℬ ⟹ openin (top_of_set S) C›)
    show "∃𝒰⊆ℬ - {{}}. T = ⋃𝒰" if "openin (top_of_set S) T" for T
      using ℬ [OF that]
      apply clarify
      by (rule_tac x="𝒰 - {{}}" in exI, auto)
  qed auto
qed

lemma Lindelof_openin:
  fixes ℱ :: "'a::second_countable_topology set set"
  assumes "⋀S. S ∈ ℱ ⟹ openin (top_of_set U) S"
  obtains ℱ' where "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
proof -
  have "⋀S. S ∈ ℱ ⟹ ∃T. open T ∧ S = U ∩ T"
    using assms by (simp add: openin_open)
  then obtain tf where tf: "⋀S. S ∈ ℱ ⟹ open (tf S) ∧ (S = U ∩ tf S)"
    by metis
  have [simp]: "⋀ℱ'. ℱ' ⊆ ℱ ⟹ ⋃ℱ' = U ∩ ⋃(tf ` ℱ')"
    using tf by fastforce
  obtain 𝒢 where "countable 𝒢 ∧ 𝒢 ⊆ tf ` ℱ" "⋃𝒢 = ⋃(tf ` ℱ)"
    using tf by (force intro: Lindelof [of "tf ` ℱ"])
  then obtain ℱ' where ℱ': "ℱ' ⊆ ℱ" "countable ℱ'" "⋃ℱ' = ⋃ℱ"
    by (clarsimp simp add: countable_subset_image)
  then show ?thesis ..
qed


subsection✐‹tag unimportant›‹Closed Maps›

lemma continuous_imp_closed_map:
  fixes f :: "'a::t2_space ⇒ 'b::t2_space"
  assumes "closedin (top_of_set S) U"
          "continuous_on S f" "f ` S = T" "compact S"
    shows "closedin (top_of_set T) (f ` U)"
  by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)

lemma closed_map_restrict:
  assumes cloU: "closedin (top_of_set (S ∩ f -` T')) U"
    and cc: "⋀U. closedin (top_of_set S) U ⟹ closedin (top_of_set T) (f ` U)"
    and "T' ⊆ T"
  shows "closedin (top_of_set T') (f ` U)"
proof -
  obtain V where "closed V" "U = S ∩ f -` T' ∩ V"
    using cloU by (auto simp: closedin_closed)
  with cc [of "S ∩ V"] ‹T' ⊆ T› show ?thesis
    by (fastforce simp add: closedin_closed)
qed

subsection✐‹tag unimportant›‹Open Maps›

lemma open_map_restrict:
  assumes opeU: "openin (top_of_set (S ∩ f -` T')) U"
    and oo: "⋀U. openin (top_of_set S) U ⟹ openin (top_of_set T) (f ` U)"
    and "T' ⊆ T"
  shows "openin (top_of_set T') (f ` U)"
proof -
  obtain V where "open V" "U = S ∩ f -` T' ∩ V"
    using opeU by (auto simp: openin_open)
  with oo [of "S ∩ V"] ‹T' ⊆ T› show ?thesis
    by (fastforce simp add: openin_open)
qed


subsection✐‹tag unimportant›‹Quotient maps›

lemma quotient_map_imp_continuous_open:
  assumes T: "f ∈ S → T"
      and ope: "⋀U. U ⊆ T
              ⟹ (openin (top_of_set S) (S ∩ f -` U) ⟷
                   openin (top_of_set T) U)"
    shows "continuous_on S f"
proof -
  have [simp]: "S ∩ f -` f ` S = S" by auto
  show ?thesis
    by (meson T continuous_on_open_gen ope openin_imp_subset)
qed

lemma quotient_map_imp_continuous_closed:
  assumes T: "f ∈ S → T"
      and ope: "⋀U. U ⊆ T
                  ⟹ (closedin (top_of_set S) (S ∩ f -` U) ⟷
                       closedin (top_of_set T) U)"
    shows "continuous_on S f"
proof -
  have [simp]: "S ∩ f -` f ` S = S" by auto
  show ?thesis
    by (meson T closedin_imp_subset continuous_on_closed_gen ope)
qed

lemma open_map_imp_quotient_map:
  assumes contf: "continuous_on S f"
      and T: "T ⊆ f ` S"
      and ope: "⋀T. openin (top_of_set S) T
                   ⟹ openin (top_of_set (f ` S)) (f ` T)"
    shows "openin (top_of_set S) (S ∩ f -` T) =
           openin (top_of_set (f ` S)) T"
proof -
  have "T = f ` (S ∩ f -` T)"
    using T by blast
  then show ?thesis
    using "ope" contf continuous_on_open by metis
qed

lemma closed_map_imp_quotient_map:
  assumes contf: "continuous_on S f"
      and T: "T ⊆ f ` S"
      and ope: "⋀T. closedin (top_of_set S) T
              ⟹ closedin (top_of_set (f ` S)) (f ` T)"
    shows "openin (top_of_set S) (S ∩ f -` T) ⟷
           openin (top_of_set (f ` S)) T"
          (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have *: "closedin (top_of_set S) (S - (S ∩ f -` T))"
    using closedin_diff by fastforce
  have [simp]: "(f ` S - f ` (S - (S ∩ f -` T))) = T"
    using T by blast
  show ?rhs
    using ope [OF *, unfolded closedin_def] by auto
next
  assume ?rhs
  with contf show ?lhs
    by (auto simp: continuous_on_open)
qed

lemma continuous_right_inverse_imp_quotient_map:
  assumes contf: "continuous_on S f" and imf: "f ∈ S → T"
      and contg: "continuous_on T g" and img: "g ∈ T → S"
      and fg [simp]: "⋀y. y ∈ T ⟹ f(g y) = y"
      and U: "U ⊆ T"
    shows "openin (top_of_set S) (S ∩ f -` U) ⟷
           openin (top_of_set T) U"
          (is "?lhs = ?rhs")
proof -
  have f: "⋀Z. openin (top_of_set (f ` S)) Z ⟹
                openin (top_of_set S) (S ∩ f -` Z)"
  and  g: "⋀Z. openin (top_of_set (g ` T)) Z ⟹
                openin (top_of_set T) (T ∩ g -` Z)"
    using contf contg by (auto simp: continuous_on_open)
  show ?thesis
  proof
    have "T ∩ g -` (g ` T ∩ (S ∩ f -` U)) = {x ∈ T. f (g x) ∈ U}"
      using imf img by blast
    also have "... = U"
      using U by auto
    finally have eq: "T ∩ g -` (g ` T ∩ (S ∩ f -` U)) = U" .
    assume ?lhs
    then have *: "openin (top_of_set (g ` T)) (g ` T ∩ (S ∩ f -` U))"
      by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self image_subset_iff_funcset)
    show ?rhs
      using g [OF *] eq by auto
  next
    assume rhs: ?rhs
    show ?lhs
      using assms continuous_openin_preimage rhs by blast
  qed
qed

lemma continuous_left_inverse_imp_quotient_map:
  assumes "continuous_on S f"
      and "continuous_on (f ` S) g"
      and  "⋀x. x ∈ S ⟹ g(f x) = x"
      and "U ⊆ f ` S"
    shows "openin (top_of_set S) (S ∩ f -` U) ⟷
           openin (top_of_set (f ` S)) U"
  using assms 
  by (intro continuous_right_inverse_imp_quotient_map) auto

lemma continuous_imp_quotient_map:
  fixes f :: "'a::t2_space ⇒ 'b::t2_space"
  assumes "continuous_on S f" "f ` S = T" "compact S" "U ⊆ T"
    shows "openin (top_of_set S) (S ∩ f -` U) ⟷
           openin (top_of_set T) U"
  by (simp add: assms closed_map_imp_quotient_map continuous_imp_closed_map)

subsection✐‹tag unimportant›‹Pasting lemmas for functions, for of casewise definitions›

subsubsection‹on open sets›

lemma pasting_lemma:
  assumes ope: "⋀i. i ∈ I ⟹ openin X (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_map(subtopology X (T i)) Y (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
      and g: "⋀x. x ∈ topspace X ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
    shows "continuous_map X Y g"
  unfolding continuous_map_openin_preimage_eq
proof (intro conjI allI impI)
  show "g ∈ topspace X → topspace Y"
    using g cont continuous_map_image_subset_topspace by fastforce
next
  fix U
  assume Y: "openin Y U"
  have T: "T i ⊆ topspace X" if "i ∈ I" for i
    using ope by (simp add: openin_subset that)
  have *: "topspace X ∩ g -` U = (⋃i ∈ I. T i ∩ f i -` U)"
    using f g T by fastforce
  have "⋀i. i ∈ I ⟹ openin X (T i ∩ f i -` U)"
    using cont unfolding continuous_map_openin_preimage_eq
    by (metis Y T inf.commute inf_absorb1 ope topspace_subtopology openin_trans_full)
  then show "openin X (topspace X ∩ g -` U)"
    by (auto simp: *)
qed

lemma pasting_lemma_exists:
  assumes X: "topspace X ⊆ (⋃i ∈ I. T i)"
      and ope: "⋀i. i ∈ I ⟹ openin X (T i)"
      and cont: "⋀i. i ∈ I ⟹ continuous_map (subtopology X (T i)) Y (f i)"
      and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    obtains g where "continuous_map X Y g" "⋀x i. ⟦i ∈ I; x ∈ topspace X ∩ T i⟧ ⟹ g x = f i x"
proof
  let ?h = "λx. f (SOME i. i ∈ I ∧ x ∈ T i) x"
  show "continuous_map X Y ?h"
    apply (rule pasting_lemma [OF ope cont])
     apply (blast intro: f)+
    by (metis (no_types, lifting) UN_E X subsetD someI_ex)
  show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x" if "i ∈ I" "x ∈ topspace X ∩ T i" for i x
    by (metis (no_types, lifting) IntD2 IntI f someI_ex that)
qed

lemma pasting_lemma_locally_finite:
  assumes fin: "⋀x. x ∈ topspace X ⟹ ∃V. openin X V ∧ x ∈ V ∧ finite {i ∈ I. T i ∩ V ≠ {}}"
    and clo: "⋀i. i ∈ I ⟹ closedin X (T i)"
    and cont:  "⋀i. i ∈ I ⟹ continuous_map(subtopology X (T i)) Y (f i)"
    and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    and g: "⋀x. x ∈ topspace X ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
  shows "continuous_map X Y g"
  unfolding continuous_map_closedin_preimage_eq
proof (intro conjI allI impI)
  show "g ∈ topspace X → topspace Y"
    using g cont continuous_map_image_subset_topspace by fastforce
next
  fix U
  assume Y: "closedin Y U"
  have T: "T i ⊆ topspace X" if "i ∈ I" for i
    using clo by (simp add: closedin_subset that)
  have *: "topspace X ∩ g -` U = (⋃i ∈ I. T i ∩ f i -` U)"
    using f g T by fastforce
  have cTf: "⋀i. i ∈ I ⟹ closedin X (T i ∩ f i -` U)"
    using cont unfolding continuous_map_closedin_preimage_eq topspace_subtopology
    by (simp add: Int_absorb1 T Y clo closedin_closed_subtopology)
  have sub: "{Z ∈ (λi. T i ∩ f i -` U) ` I. Z ∩ V ≠ {}}
           ⊆ (λi. T i ∩ f i -` U) ` {i ∈ I. T i ∩ V ≠ {}}" for V
    by auto
  have 1: "(⋃i∈I. T i ∩ f i -` U) ⊆ topspace X"
    using T by blast
  then have "locally_finite_in X ((λi. T i ∩ f i -` U) ` I)"
    unfolding locally_finite_in_def
    using finite_subset [OF sub] fin by force
  then show "closedin X (topspace X ∩ g -` U)"
    by (smt (verit, best) * cTf closedin_locally_finite_Union image_iff)
qed

subsubsection‹Likewise on closed sets, with a finiteness assumption›

lemma pasting_lemma_closed:
  assumes fin: "finite I"
    and clo: "⋀i. i ∈ I ⟹ closedin X (T i)"
    and cont:  "⋀i. i ∈ I ⟹ continuous_map(subtopology X (T i)) Y (f i)"
    and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    and g: "⋀x. x ∈ topspace X ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
  shows "continuous_map X Y g"
  using pasting_lemma_locally_finite [OF _ clo cont f g] fin by auto

lemma pasting_lemma_exists_locally_finite:
  assumes fin: "⋀x. x ∈ topspace X ⟹ ∃V. openin X V ∧ x ∈ V ∧ finite {i ∈ I. T i ∩ V ≠ {}}"
    and X: "topspace X ⊆ ⋃(T ` I)"
    and clo: "⋀i. i ∈ I ⟹ closedin X (T i)"
    and cont:  "⋀i. i ∈ I ⟹ continuous_map(subtopology X (T i)) Y (f i)"
    and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
    and g: "⋀x. x ∈ topspace X ⟹ ∃j. j ∈ I ∧ x ∈ T j ∧ g x = f j x"
  obtains g where "continuous_map X Y g" "⋀x i. ⟦i ∈ I; x ∈ topspace X ∩ T i⟧ ⟹ g x = f i x"
proof
  show "continuous_map X Y (λx. f(@i. i ∈ I ∧ x ∈ T i) x)"
    apply (rule pasting_lemma_locally_finite [OF fin])
        apply (blast intro: assms)+
    by (metis (no_types, lifting) UN_E X set_rev_mp someI_ex)
next
  fix x i
  assume "i ∈ I" and "x ∈ topspace X ∩ T i"
  then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
    by (metis (mono_tags, lifting) IntE IntI f someI2)
qed

lemma pasting_lemma_exists_closed:
  assumes fin: "finite I"
    and X: "topspace X ⊆ ⋃(T ` I)"
    and clo: "⋀i. i ∈ I ⟹ closedin X (T i)"
    and cont:  "⋀i. i ∈ I ⟹ continuous_map(subtopology X (T i)) Y (f i)"
    and f: "⋀i j x. ⟦i ∈ I; j ∈ I; x ∈ topspace X ∩ T i ∩ T j⟧ ⟹ f i x = f j x"
  obtains g where "continuous_map X Y g" "⋀x i. ⟦i ∈ I; x ∈ topspace X ∩ T i⟧ ⟹ g x = f i x"
proof
  show "continuous_map X Y (λx. f (SOME i. i ∈ I ∧ x ∈ T i) x)"
    apply (rule pasting_lemma_closed [OF ‹finite I› clo cont])
     apply (blast intro: f)+
    by (metis (mono_tags, lifting) UN_iff X someI_ex subset_iff)
next
  fix x i
  assume "i ∈ I" "x ∈ topspace X ∩ T i"
  then show "f (SOME i. i ∈ I ∧ x ∈ T i) x = f i x"
    by (metis (no_types, lifting) IntD2 IntI f someI_ex)
qed

lemma continuous_map_cases:
  assumes f: "continuous_map (subtopology X (X closure_of {x. P x})) Y f"
      and g: "continuous_map (subtopology X (X closure_of {x. ¬ P x})) Y g"
      and fg: "⋀x. x ∈ X frontier_of {x. P x} ⟹ f x = g x"
  shows "continuous_map X Y (λx. if P x then f x else g x)"
proof (rule pasting_lemma_closed)
  let ?f = "λb. if b then f else g"
  let ?g = "λx. if P x then f x else g x"
  let ?T = "λb. if b then X closure_of {x. P x} else X closure_of {x. ~P x}"
  show "finite {True,False}" by auto
  have eq: "topspace X - Collect P = topspace X ∩ {x. ¬ P x}"
    by blast
  show "?f i x = ?f j x"
    if "i ∈ {True,False}" "j ∈ {True,False}" and x: "x ∈ topspace X ∩ ?T i ∩ ?T j" for i j x
  proof -
    have "f x = g x" if "i" "¬ j"
      by (smt (verit, best) Diff_Diff_Int closure_of_interior_of closure_of_restrict eq fg 
          frontier_of_closures interior_of_complement that x)
    moreover
    have "g x = f x"
      if "x ∈ X closure_of {x. ¬ P x}" "x ∈ X closure_of Collect P" "¬ i" "j" for x
      by (metis IntI closure_of_restrict eq fg frontier_of_closures that)
    ultimately show ?thesis
      using that by (auto simp flip: closure_of_restrict)
  qed
  show "∃j. j ∈ {True,False} ∧ x ∈ ?T j ∧ (if P x then f x else g x) = ?f j x"
    if "x ∈ topspace X" for x
    by simp (metis in_closure_of mem_Collect_eq that)
qed (auto simp: f g)

lemma continuous_map_cases_alt:
  assumes f: "continuous_map (subtopology X (X closure_of {x ∈ topspace X. P x})) Y f"
      and g: "continuous_map (subtopology X (X closure_of {x ∈ topspace X. ~P x})) Y g"
      and fg: "⋀x. x ∈ X frontier_of {x ∈ topspace X. P x} ⟹ f x = g x"
    shows "continuous_map X Y (λx. if P x then f x else g x)"
  apply (rule continuous_map_cases)
  using assms
    apply (simp_all add: Collect_conj_eq closure_of_restrict [symmetric] frontier_of_restrict [symmetric])
  done

lemma continuous_map_cases_function:
  assumes contp: "continuous_map X Z p"
    and contf: "continuous_map (subtopology X {x ∈ topspace X. p x ∈ Z closure_of U}) Y f"
    and contg: "continuous_map (subtopology X {x ∈ topspace X. p x ∈ Z closure_of (topspace Z - U)}) Y g"
    and fg: "⋀x. ⟦x ∈ topspace X; p x ∈ Z frontier_of U⟧ ⟹ f x = g x"
  shows "continuous_map X Y (λx. if p x ∈ U then f x else g x)"
proof (rule continuous_map_cases_alt)
  show "continuous_map (subtopology X (X closure_of {x ∈ topspace X. p x ∈ U})) Y f"
  proof (rule continuous_map_from_subtopology_mono)
    let ?T = "{x ∈ topspace X. p x ∈ Z closure_of U}"
    show "continuous_map (subtopology X ?T) Y f"
      by (simp add: contf)
    show "X closure_of {x ∈ topspace X. p x ∈ U} ⊆ ?T"
      by (rule continuous_map_closure_preimage_subset [OF contp])
  qed
  show "continuous_map (subtopology X (X closure_of {x ∈ topspace X. p x ∉ U})) Y g"
  proof (rule continuous_map_from_subtopology_mono)
    let ?T = "{x ∈ topspace X. p x ∈ Z closure_of (topspace Z - U)}"
    show "continuous_map (subtopology X ?T) Y g"
      by (simp add: contg)
    have "X closure_of {x ∈ topspace X. p x ∉ U} ⊆ X closure_of {x ∈ topspace X. p x ∈ topspace Z - U}"
      by (smt (verit) Collect_mono_iff DiffI closure_of_mono continuous_map contp image_subset_iff)
    then show "X closure_of {x ∈ topspace X. p x ∉ U} ⊆ ?T"
      by (rule order_trans [OF _ continuous_map_closure_preimage_subset [OF contp]])
  qed
next
  show "f x = g x" if "x ∈ X frontier_of {x ∈ topspace X. p x ∈ U}" for x
    using that continuous_map_frontier_frontier_preimage_subset [OF contp, of U] fg by blast
qed

subsection ‹Retractions›

definition✐‹tag important› retraction :: "('a::topological_space) set ⇒ 'a set ⇒ ('a ⇒ 'a) ⇒ bool"
where "retraction S T r ⟷
  T ⊆ S ∧ continuous_on S r ∧ r ∈ S → T ∧ (∀x∈T. r x = x)"

definition✐‹tag important› retract_of (infixl "retract'_of" 50) where
"T retract_of S  ⟷  (∃r. retraction S T r)"

lemma retraction_idempotent: "retraction S T r ⟹ x ∈ S ⟹  r (r x) = r x"
  unfolding retraction_def by auto

text ‹Preservation of fixpoints under (more general notion of) retraction›

lemma invertible_fixpoint_property:
  fixes S :: "'a::topological_space set"
    and T :: "'b::topological_space set"
  assumes contt: "continuous_on T i"
    and "i ∈ T → S"
    and contr: "continuous_on S r"
    and "r ∈ S → T"
    and ri: "⋀y. y ∈ T ⟹ r (i y) = y"
    and FP: "⋀f. ⟦continuous_on S f; f ∈ S → S⟧ ⟹ ∃x∈S. f x = x"
    and contg: "continuous_on T g"
    and "g ∈ T → T"
  obtains y where "y ∈ T" and "g y = y"
proof -
  have "∃x∈S. (i ∘ g ∘ r) x = x"
  proof (rule FP)
    show "continuous_on S (i ∘ g ∘ r)"
      by (metis assms(4) assms(8) contg continuous_on_compose continuous_on_subset contr contt funcset_image)
    show "(i ∘ g ∘ r) ∈ S → S"
      using assms(2,4,8) by force
  qed
  then obtain x where x: "x ∈ S" "(i ∘ g ∘ r) x = x" ..
  then have *: "g (r x) ∈ T"
    using assms(4,8) by auto
  have "r ((i ∘ g ∘ r) x) = r x"
    using x by auto
  then show ?thesis
    using "*" ri that by auto
qed

lemma homeomorphic_fixpoint_property:
  fixes S :: "'a::topological_space set"
    and T :: "'b::topological_space set"
  assumes "S homeomorphic T"
  shows "(∀f. continuous_on S f ∧ f ∈ S → S ⟶ (∃x∈S. f x = x)) ⟷
         (∀g. continuous_on T g ∧ g ∈ T → T ⟶ (∃y∈T. g y = y))"
         (is "?lhs = ?rhs")
proof -
  obtain r i where r:
      "∀x∈S. i (r x) = x" "r ` S = T" "continuous_on S r"
      "∀y∈T. r (i y) = y" "i ` T = S" "continuous_on T i"
    using assms unfolding homeomorphic_def homeomorphism_def  by blast
  show ?thesis
  proof
    assume ?lhs
    with r show ?rhs
      by (smt (verit, del_insts) Pi_iff image_eqI invertible_fixpoint_property[of T i S r])
  next
    assume ?rhs
    with r show ?lhs
      by (smt (verit, del_insts) Pi_iff image_eqI invertible_fixpoint_property[of S r T i])
  qed
qed

lemma retract_fixpoint_property:
  fixes f :: "'a::topological_space ⇒ 'b::topological_space"
    and S :: "'a set"
  assumes "T retract_of S"
    and FP: "⋀f. ⟦continuous_on S f; f ∈ S → S⟧ ⟹ ∃x∈S. f x = x"
    and contg: "continuous_on T g"
    and "g ∈ T → T"
  obtains y where "y ∈ T" and "g y = y"
proof -
  obtain h where "retraction S T h"
    using assms(1) unfolding retract_of_def ..
  then show ?thesis
    unfolding retraction_def
    using invertible_fixpoint_property[OF continuous_on_id _ _ _ _ FP]
    by (smt (verit, del_insts) Pi_iff assms(4) contg subsetD that)
qed

lemma retraction:
  "retraction S T r ⟷ T ⊆ S ∧ continuous_on S r ∧ r ` S = T ∧ (∀x ∈ T. r x = x)"
  by (force simp: retraction_def simp flip: image_subset_iff_funcset)

lemma retractionE: ― ‹yields properties normalized wrt. simp -- less likely to loop›
  assumes "retraction S T r"
  obtains "T = r ` S" "r ∈ S → S" "continuous_on S r" "⋀x. x ∈ S ⟹ r (r x) = r x"
proof (rule that)
  from retraction [of S T r] assms
  have "T ⊆ S" "continuous_on S r" "r ` S = T" and "∀x ∈ T. r x = x"
    by simp_all
  then show  "r ∈ S → S" "continuous_on S r"
    by auto
  then show "T = r ` S"
    using ‹r ` S = T› by blast
  from ‹∀x ∈ T. r x = x› have "r x = x" if "x ∈ T" for x
    using that by simp
  with ‹r ` S = T› show "r (r x) = r x" if "x ∈ S" for x
    using that by auto
qed

lemma retract_ofE: ― ‹yields properties normalized wrt. simp -- less likely to loop›
  assumes "T retract_of S"
  obtains r where "T = r ` S" "r ∈ S → S" "continuous_on S r" "⋀x. x ∈ S ⟹ r (r x) = r x"
proof -
  from assms obtain r where "retraction S T r"
    by (auto simp add: retract_of_def)
  with that show thesis
    by (auto elim: retractionE)
qed

lemma retract_of_imp_extensible:
  assumes "S retract_of T" and "continuous_on S f" and "f ∈ S → U"
  obtains g where "continuous_on T g" "g ∈ T → U" "⋀x. x ∈ S ⟹ g x = f x"
proof -
  from ‹S retract_of T› obtain r where "retraction T S r"
    by (auto simp add: retract_of_def)
  then have "continuous_on T (f ∘ r)"
      by (metis assms(2) continuous_on_compose retraction)
  then show thesis
    by (smt (verit, best) Pi_iff ‹retraction T S r› assms(3) comp_apply retraction_def that)
qed

lemma idempotent_imp_retraction:
  assumes "continuous_on S f" and "f ∈ S → S" and "⋀x. x ∈ S ⟹ f(f x) = f x"
    shows "retraction S (f ` S) f"
  by (simp add: assms funcset_image retraction)

lemma retraction_subset:
  assumes "retraction S T r" and "T ⊆ S'" and "S' ⊆ S"
  shows "retraction S' T r"
  unfolding retraction_def
  by (metis assms continuous_on_subset image_mono image_subset_iff_funcset retraction)

lemma retract_of_subset:
  assumes "T retract_of S" and "T ⊆ S'" and "S' ⊆ S"
    shows "T retract_of S'"
by (meson assms retract_of_def retraction_subset)

lemma retraction_refl [simp]: "retraction S S (λx. x)"
by (simp add: retraction)

lemma retract_of_refl [iff]: "S retract_of S"
  unfolding retract_of_def retraction_def
  using continuous_on_id by blast

lemma retract_of_imp_subset:
   "S retract_of T ⟹ S ⊆ T"
by (simp add: retract_of_def retraction_def)

lemma retract_of_empty [simp]:
     "({} retract_of S) ⟷ S = {}"  "(S retract_of {}) ⟷ S = {}"
by (auto simp: retract_of_def retraction_def)

lemma retract_of_singleton [iff]: "({x} retract_of S) ⟷ x ∈ S"
  unfolding retract_of_def retraction_def by force

lemma retraction_comp:
   "⟦retraction S T f; retraction T U g⟧ ⟹ retraction S U (g ∘ f)"
  by (smt (verit, best) comp_apply continuous_on_compose image_comp retraction subset_iff)

lemma retract_of_trans [trans]:
  assumes "S retract_of T" and "T retract_of U"
    shows "S retract_of U"
using assms by (auto simp: retract_of_def intro: retraction_comp)

lemma closedin_retract:
  fixes S :: "'a :: t2_space set"
  assumes "S retract_of T"
    shows "closedin (top_of_set T) S"
proof -
  obtain r where r: "S ⊆ T" "continuous_on T r" "r ∈ T → S" "⋀x. x ∈ S ⟹ r x = x"
    using assms by (auto simp: retract_of_def retraction_def)
  have "S = {x∈T. x = r x}"
    using r by force
  also have "… = T ∩ ((λx. (x, r x)) -` ({y. ∃x. y = (x, x)}))"
    unfolding vimage_def mem_Times_iff fst_conv snd_conv
    using r
    by auto
  also have "closedin (top_of_set T) …"
    by (rule continuous_closedin_preimage) (auto intro!: closed_diagonal continuous_on_Pair r)
  finally show ?thesis .
qed

lemma closedin_self [simp]: "closedin (top_of_set S) S"
  by simp

lemma retract_of_closed:
    fixes S :: "'a :: t2_space set"
    shows "⟦closed T; S retract_of T⟧ ⟹ closed S"
  by (metis closedin_retract closedin_closed_eq)

lemma retract_of_compact:
     "⟦compact T; S retract_of T⟧ ⟹ compact S"
  by (metis compact_continuous_image retract_of_def retraction)

lemma retract_of_connected:
    "⟦connected T; S retract_of T⟧ ⟹ connected S"
  by (metis Topological_Spaces.connected_continuous_image retract_of_def retraction)

lemma retraction_openin_vimage_iff:
  assumes r: "retraction S T r" and "U ⊆ T"
  shows "openin (top_of_set S) (S ∩ r -` U) ⟷ openin (top_of_set T) U" (is "?lhs = ?rhs")
proof
  show "?lhs ⟹ ?rhs"
    using r
    by (smt (verit, del_insts) assms(2) continuous_right_inverse_imp_quotient_map image_subset_iff_funcset r retractionE retraction_def retraction_subset)
  show "?rhs ⟹ ?lhs"
    by (metis continuous_on_open r retraction)
qed

lemma retract_of_Times:
   "⟦S retract_of S'; T retract_of T'⟧ ⟹ (S × T) retract_of (S' × T')"
apply (simp add: retract_of_def retraction_def Sigma_mono, clarify)
apply (rename_tac f g)
apply (rule_tac x="λz. ((f ∘ fst) z, (g ∘ snd) z)" in exI)
apply (rule conjI continuous_intros | erule continuous_on_subset | force)+
done

subsection‹Retractions on a topological space›

definition retract_of_space :: "'a set ⇒ 'a topology ⇒ bool" (infix "retract'_of'_space" 50)
  where "S retract_of_space X
         ≡ S ⊆ topspace X ∧ (∃r. continuous_map X (subtopology X S) r ∧ (∀x ∈ S. r x = x))"

lemma retract_of_space_retraction_maps:
   "S retract_of_space X ⟷ S ⊆ topspace X ∧ (∃r. retraction_maps X (subtopology X S) r id)"
  by (auto simp: retract_of_space_def retraction_maps_def)

lemma retract_of_space_section_map:
   "S retract_of_space X ⟷ S ⊆ topspace X ∧ section_map (subtopology X S) X id"
  unfolding retract_of_space_def retraction_maps_def section_map_def
  by (auto simp: continuous_map_from_subtopology)

lemma retract_of_space_imp_subset:
   "S retract_of_space X ⟹ S ⊆ topspace X"
  by (simp add: retract_of_space_def)

lemma retract_of_space_topspace:
   "topspace X retract_of_space X"
  using retract_of_space_def by force

lemma retract_of_space_empty [simp]:
   "{} retract_of_space X ⟷ X = trivial_topology"
  by (auto simp: retract_of_space_def)

lemma retract_of_space_singleton [simp]:
  "{a} retract_of_space X ⟷ a ∈ topspace X"
proof -
  have "continuous_map X (subtopology X {a}) (λx. a) ∧ (λx. a) a = a" if "a ∈ topspace X"
    using that by simp
  then show ?thesis
    by (force simp: retract_of_space_def)
qed

lemma retract_of_space_trans:
  assumes "S retract_of_space X"  "T retract_of_space subtopology X S"
  shows "T retract_of_space X"
  using assms
  apply (simp add: retract_of_space_retraction_maps)
  by (metis id_comp inf.absorb_iff2 retraction_maps_compose subtopology_subtopology)

lemma retract_of_space_subtopology:
  assumes "S retract_of_space X"  "S ⊆ U"
  shows "S retract_of_space subtopology X U"
  using assms
  apply (clarsimp simp: retract_of_space_def)
  by (metis continuous_map_from_subtopology inf.absorb2 subtopology_subtopology)

lemma retract_of_space_clopen:
  assumes "openin X S" "closedin X S" "S = {} ⟹ X = trivial_topology"
  shows "S retract_of_space X"
proof (cases "S = {}")
  case False
  then obtain a where "a ∈ S"
    by blast
  show ?thesis
    unfolding retract_of_space_def
  proof (intro exI conjI)
    show "S ⊆ topspace X"
      by (simp add: assms closedin_subset)
    have "continuous_map X X (λx. if x ∈ S then x else a)"
    proof (rule continuous_map_cases)
      show "continuous_map (subtopology X (X closure_of {x. x ∈ S})) X (λx. x)"
        by (simp add: continuous_map_from_subtopology)
      show "continuous_map (subtopology X (X closure_of {x. x ∉ S})) X (λx. a)"
        using ‹S ⊆ topspace X› ‹a ∈ S› by force
      show "x = a" if "x ∈ X frontier_of {x. x ∈ S}" for x
        using assms that clopenin_eq_frontier_of by fastforce
    qed
    then show "continuous_map X (subtopology X S) (λx. if x ∈ S then x else a)"
      using ‹S ⊆ topspace X› ‹a ∈ S›  by (auto simp: continuous_map_in_subtopology)
  qed auto
qed (use assms in auto)

lemma retract_of_space_disjoint_union:
  assumes "openin X S" "openin X T" and ST: "disjnt S T" "S ∪ T = topspace X" and "S = {} ⟹ X = trivial_topology"
  shows "S retract_of_space X"
proof (rule retract_of_space_clopen)
  have "S ∩ T = {}"
    by (meson ST disjnt_def)
  then have "S = topspace X - T"
    using ST by auto
  then show "closedin X S"
    using ‹openin X T› by blast
qed (auto simp: assms)

lemma retraction_maps_section_image1:
  assumes "retraction_maps X Y r s"
  shows "s ` (topspace Y) retract_of_space X"
  unfolding retract_of_space_section_map
proof
  show "s ` topspace Y ⊆ topspace X"
    using assms continuous_map_image_subset_topspace retraction_maps_def by blast
  show "section_map (subtopology X (s ` topspace Y)) X id"
    unfolding section_map_def
    using assms retraction_maps_to_retract_maps by blast
qed

lemma retraction_maps_section_image2:
   "retraction_maps X Y r s
        ⟹ subtopology X (s ` (topspace Y)) homeomorphic_space Y"
  using embedding_map_imp_homeomorphic_space homeomorphic_space_sym section_imp_embedding_map
        section_map_def by blast

lemma hereditary_imp_retractive_property:
  assumes "⋀X S. P X ⟹ P(subtopology X S)" 
          "⋀X X'. X homeomorphic_space X' ⟹ (P X ⟷ Q X')"
  assumes "retraction_map X X' r" "P X"
  shows "Q X'"
  by (meson assms retraction_map_def retraction_maps_section_image2)

subsection‹Paths and path-connectedness›

definition pathin :: "'a topology ⇒ (real ⇒ 'a) ⇒ bool" where
   "pathin X g ≡ continuous_map (subtopology euclideanreal {0..1}) X g"

lemma pathin_compose:
     "⟦pathin X g; continuous_map X Y f⟧ ⟹ pathin Y (f ∘ g)"
   by (simp add: continuous_map_compose pathin_def)

lemma pathin_subtopology:
     "pathin (subtopology X S) g ⟷ pathin X g ∧ (∀x ∈ {0..1}. g x ∈ S)"
  by (auto simp: pathin_def continuous_map_in_subtopology)

lemma pathin_const [simp]: "pathin X (λx. a) ⟷ a ∈ topspace X"
  using pathin_subtopology by (fastforce simp add: pathin_def)

lemma path_start_in_topspace: "pathin X g ⟹ g 0 ∈ topspace X"
  by (force simp: pathin_def continuous_map)

lemma path_finish_in_topspace: "pathin X g ⟹ g 1 ∈ topspace X"
  by (force simp: pathin_def continuous_map)

lemma path_image_subset_topspace: "pathin X g ⟹ g ∈ ({0..1}) → topspace X"
  by (force simp: pathin_def continuous_map)

definition path_connected_space :: "'a topology ⇒ bool"
  where "path_connected_space X ≡ ∀x ∈ topspace X. ∀ y ∈ topspace X. ∃g. pathin X g ∧ g 0 = x ∧ g 1 = y"

definition path_connectedin :: "'a topology ⇒ 'a set ⇒ bool"
  where "path_connectedin X S ≡ S ⊆ topspace X ∧ path_connected_space(subtopology X S)"

lemma path_connectedin_absolute [simp]:
     "path_connectedin (subtopology X S) S ⟷ path_connectedin X S"
  by (simp add: path_connectedin_def subtopology_subtopology)

lemma path_connectedin_subset_topspace:
     "path_connectedin X S ⟹ S ⊆ topspace X"
  by (simp add: path_connectedin_def)

lemma path_connectedin_subtopology:
     "path_connectedin (subtopology X S) T ⟷ path_connectedin X T ∧ T ⊆ S"
  by (auto simp: path_connectedin_def subtopology_subtopology inf.absorb2)

lemma path_connectedin:
     "path_connectedin X S ⟷
        S ⊆ topspace X ∧
        (∀x ∈ S. ∀y ∈ S. ∃g. pathin X g ∧ g ∈ {0..1} → S ∧ g 0 = x ∧ g 1 = y)"
  unfolding path_connectedin_def path_connected_space_def pathin_def continuous_map_in_subtopology
  by (intro conj_cong refl ball_cong) (simp_all add: inf.absorb_iff2 flip: image_subset_iff_funcset)

lemma path_connectedin_topspace:
     "path_connectedin X (topspace X) ⟷ path_connected_space X"
  by (simp add: path_connectedin_def)

lemma path_connected_imp_connected_space:
  assumes "path_connected_space X"
  shows "connected_space X"
proof -
  have *: "∃S. connectedin X S ∧ g 0 ∈ S ∧ g 1 ∈ S" if "pathin X g" for g
  proof (intro exI conjI)
    have "continuous_map (subtopology euclideanreal {0..1}) X g"
      using connectedin_absolute that by (simp add: pathin_def)
    then show "connectedin X (g ` {0..1})"
      by (rule connectedin_continuous_map_image) auto
  qed auto
  show ?thesis
    using assms
    by (auto intro: * simp add: path_connected_space_def connected_space_subconnected Ball_def)
qed

lemma path_connectedin_imp_connectedin:
     "path_connectedin X S ⟹ connectedin X S"
  by (simp add: connectedin_def path_connected_imp_connected_space path_connectedin_def)

lemma path_connected_space_topspace_empty:
     "path_connected_space trivial_topology"
  by (simp add: path_connected_space_def)

lemma path_connectedin_empty [simp]: "path_connectedin X {}"
  by (simp add: path_connectedin)

lemma path_connectedin_singleton [simp]: "path_connectedin X {a} ⟷ a ∈ topspace X"
proof
  show "path_connectedin X {a} ⟹ a ∈ topspace X"
    by (simp add: path_connectedin)
  show "a ∈ topspace X ⟹ path_connectedin X {a}"
    unfolding path_connectedin
    using pathin_const by fastforce
qed

lemma path_connectedin_continuous_map_image:
  assumes f: "continuous_map X Y f" and S: "path_connectedin X S"
  shows "path_connectedin Y (f ` S)"
proof -
  have fX: "f ∈ (topspace X) → topspace Y"
    using continuous_map_def f by fastforce
  show ?thesis
    unfolding path_connectedin
  proof (intro conjI ballI; clarify?)
    fix x
    assume "x ∈ S"
    show "f x ∈ topspace Y"
      using S ‹x ∈ S› fX path_connectedin_subset_topspace by fastforce
  next
    fix x y
    assume "x ∈ S" and "y ∈ S"
    then obtain g where g: "pathin X g" "g ∈ {0..1} → S" "g 0 = x" "g 1 = y"
      using S  by (force simp: path_connectedin)
    show "∃g. pathin Y g ∧ g ∈ {0..1} → f ` S ∧ g 0 = f x ∧ g 1 = f y"
    proof (intro exI conjI)
      show "pathin Y (f ∘ g)"
        using ‹pathin X g› f pathin_compose by auto
    qed (use g in auto)
  qed
qed

lemma path_connectedin_discrete_topology:
  "path_connectedin (discrete_topology U) S ⟷ S ⊆ U ∧ (∃a. S ⊆ {a})" (is "?lhs = ?rhs")
proof
  show "?lhs ⟹ ?rhs"
    by (meson connectedin_discrete_topology path_connectedin_imp_connectedin)
  show "?rhs ⟹ ?lhs"
    using subset_singletonD by fastforce
qed

lemma path_connected_space_discrete_topology:
   "path_connected_space (discrete_topology U) ⟷ (∃a. U ⊆ {a})"
  by (metis path_connectedin_discrete_topology path_connectedin_topspace path_connected_space_topspace_empty
            subset_singletonD topspace_discrete_topology)


lemma homeomorphic_path_connected_space_imp:
     "⟦path_connected_space X; X homeomorphic_space Y⟧ ⟹ path_connected_space Y"
  unfolding homeomorphic_space_def homeomorphic_maps_def
  by (smt (verit, ccfv_threshold) homeomorphic_imp_surjective_map homeomorphic_maps_def homeomorphic_maps_map path_connectedin_continuous_map_image path_connectedin_topspace)

lemma homeomorphic_path_connected_space:
   "X homeomorphic_space Y ⟹ path_connected_space X ⟷ path_connected_space Y"
  by (meson homeomorphic_path_connected_space_imp homeomorphic_space_sym)

lemma homeomorphic_map_path_connectedness:
  assumes "homeomorphic_map X Y f" "U ⊆ topspace X"
  shows "path_connectedin Y (f ` U) ⟷ path_connectedin X U"
  unfolding path_connectedin_def
proof (intro conj_cong homeomorphic_path_connected_space)
  show "f ` U ⊆ topspace Y ⟷ (U ⊆ topspace X)"
    using assms homeomorphic_imp_surjective_map by blast
next
  assume "U ⊆ topspace X"
  show "subtopology Y (f ` U) homeomorphic_space subtopology X U"
    using assms unfolding homeomorphic_eq_everything_map
    by (metis (no_types, opaque_lifting) assms homeomorphic_map_subtopologies homeomorphic_space homeomorphic_space_sym image_mono inf.absorb_iff2)
qed

lemma homeomorphic_map_path_connectedness_eq:
   "homeomorphic_map X Y f ⟹ path_connectedin X U ⟷ U ⊆ topspace X ∧ path_connectedin Y (f ` U)"
  by (meson homeomorphic_map_path_connectedness path_connectedin_def)

subsection‹Connected components›

definition connected_component_of :: "'a topology ⇒ 'a ⇒ 'a ⇒ bool"
  where "connected_component_of X x y ≡
        ∃T. connectedin X T ∧ x ∈ T ∧ y ∈ T"

abbreviation connected_component_of_set
  where "connected_component_of_set X x ≡ Collect (connected_component_of X x)"

definition connected_components_of :: "'a topology ⇒ ('a set) set"
  where "connected_components_of X ≡ connected_component_of_set X ` topspace X"

lemma connected_component_in_topspace:
   "connected_component_of X x y ⟹ x ∈ topspace X ∧ y ∈ topspace X"
  by (meson connected_component_of_def connectedin_subset_topspace in_mono)

lemma connected_component_of_refl:
   "connected_component_of X x x ⟷ x ∈ topspace X"
  by (meson connected_component_in_topspace connected_component_of_def connectedin_sing insertI1)

lemma connected_component_of_sym:
   "connected_component_of X x y ⟷ connected_component_of X y x"
  by (meson connected_component_of_def)

lemma connected_component_of_trans:
   "⟦connected_component_of X x y; connected_component_of X y z⟧
        ⟹ connected_component_of X x z"
  unfolding connected_component_of_def
  using connectedin_Un by blast

lemma connected_component_of_mono:
   "⟦connected_component_of (subtopology X S) x y; S ⊆ T⟧
        ⟹ connected_component_of (subtopology X T) x y"
  by (metis connected_component_of_def connectedin_subtopology inf.absorb_iff2 subtopology_subtopology)

lemma connected_component_of_set:
   "connected_component_of_set X x = {y. ∃T. connectedin X T ∧ x ∈ T ∧ y ∈ T}"
  by (meson connected_component_of_def)

lemma connected_component_of_subset_topspace:
   "connected_component_of_set X x ⊆ topspace X"
  using connected_component_in_topspace by force

lemma connected_component_of_eq_empty:
   "connected_component_of_set X x = {} ⟷ (x ∉ topspace X)"
  using connected_component_in_topspace connected_component_of_refl by fastforce

lemma connected_space_iff_connected_component:
   "connected_space X ⟷ (∀x ∈ topspace X. ∀y ∈ topspace X. connected_component_of X x y)"
  by (simp add: connected_component_of_def connected_space_subconnected)

lemma connected_space_imp_connected_component_of:
   "⟦connected_space X; a ∈ topspace X; b ∈ topspace X⟧
    ⟹ connected_component_of X a b"
  by (simp add: connected_space_iff_connected_component)

lemma connected_space_connected_component_set:
   "connected_space X ⟷ (∀x ∈ topspace X. connected_component_of_set X x = topspace X)"
  using connected_component_of_subset_topspace connected_space_iff_connected_component by fastforce

lemma connected_component_of_maximal:
   "⟦connectedin X S; x ∈ S⟧ ⟹ S ⊆ connected_component_of_set X x"
  by (meson Ball_Collect connected_component_of_def)

lemma connected_component_of_equiv:
   "connected_component_of X x y ⟷
    x ∈ topspace X ∧ y ∈ topspace X ∧ connected_component_of X x = connected_component_of X y"
  apply (simp add: connected_component_in_topspace fun_eq_iff)
  by (meson connected_component_of_refl connected_component_of_sym connected_component_of_trans)

lemma connected_component_of_disjoint:
   "disjnt (connected_component_of_set X x) (connected_component_of_set X y)
    ⟷ ~(connected_component_of X x y)"
  using connected_component_of_equiv unfolding disjnt_iff by force

lemma connected_component_of_eq:
   "connected_component_of X x = connected_component_of X y ⟷
        (x ∉ topspace X) ∧ (y ∉ topspace X) ∨
        x ∈ topspace X ∧ y ∈ topspace X ∧
        connected_component_of X x y"
  by (metis Collect_empty_eq_bot connected_component_of_eq_empty connected_component_of_equiv)

lemma connectedin_connected_component_of:
   "connectedin X (connected_component_of_set X x)"
proof -
  have "connected_component_of_set X x = ⋃ {T. connectedin X T ∧ x ∈ T}"
    by (auto simp: connected_component_of_def)
  then show ?thesis
    by (metis (no_types, lifting) InterI connectedin_Union emptyE mem_Collect_eq)
qed


lemma Union_connected_components_of:
   "⋃(connected_components_of X) = topspace X"
  unfolding connected_components_of_def
  using connected_component_in_topspace connected_component_of_refl by fastforce

lemma connected_components_of_maximal:
   "⟦C ∈ connected_components_of X; connectedin X S; ~disjnt C S⟧ ⟹ S ⊆ C"
  unfolding connected_components_of_def disjnt_def
  apply clarify
  by (metis Int_emptyI connected_component_of_def connected_component_of_trans mem_Collect_eq)

lemma pairwise_disjoint_connected_components_of:
   "pairwise disjnt (connected_components_of X)"
  unfolding connected_components_of_def pairwise_def
  by (smt (verit, best) connected_component_of_disjoint connected_component_of_eq imageE)

lemma complement_connected_components_of_Union:
   "C ∈ connected_components_of X
      ⟹ topspace X - C = ⋃ (connected_components_of X - {C})"
  by (metis Union_connected_components_of bot.extremum ccpo_Sup_singleton diff_Union_pairwise_disjoint
      insert_subset pairwise_disjoint_connected_components_of)

lemma nonempty_connected_components_of:
   "C ∈ connected_components_of X ⟹ C ≠ {}"
  unfolding connected_components_of_def
  by (metis (no_types, lifting) connected_component_of_eq_empty imageE)

lemma connected_components_of_subset:
   "C ∈ connected_components_of X ⟹ C ⊆ topspace X"
  using Union_connected_components_of by fastforce

lemma connectedin_connected_components_of:
  assumes "C ∈ connected_components_of X"
  shows "connectedin X C"
proof -
  have "C ∈ connected_component_of_set X ` topspace X"
    using assms connected_components_of_def by blast
then show ?thesis
  using connectedin_connected_component_of by fastforce
qed

lemma connected_component_in_connected_components_of:
  "connected_component_of_set X a ∈ connected_components_of X ⟷ a ∈ topspace X"
  by (metis (no_types, lifting) connected_component_of_eq_empty connected_components_of_def image_iff)

lemma connected_space_iff_components_eq:
   "connected_space X ⟷ (∀C ∈ connected_components_of X. ∀C' ∈ connected_components_of X. C = C')"
          (is "?lhs = ?rhs")
proof
  show "?lhs ⟹ ?rhs"
    by (simp add: connected_components_of_def connected_space_connected_component_set)
  show "?rhs ⟹ ?lhs"
    by (metis Union_connected_components_of Union_iff connected_space_subconnected connectedin_connected_components_of)
qed

lemma connected_components_of_eq_empty:
   "connected_components_of X = {} ⟷ X = trivial_topology"
  by (simp add: connected_components_of_def)

lemma connected_components_of_empty_space:
   "connected_components_of trivial_topology = {}"
  by (simp add: connected_components_of_eq_empty)

lemma connected_components_of_subset_sing:
   "connected_components_of X ⊆ {S} ⟷ connected_space X ∧ (X = trivial_topology ∨ topspace X = S)"
proof (cases "X = trivial_topology")
  case True
  then show ?thesis
    by (simp add: connected_components_of_empty_space)
next
  case False
  then have "⟦connected_components_of X ⊆ {S}⟧ ⟹ topspace X = S"
    using Union_connected_components_of connected_components_of_eq_empty by fastforce
  with False show ?thesis
    unfolding connected_components_of_def
    by (metis connected_space_connected_component_set empty_iff image_subset_iff insert_iff)
qed

lemma connected_space_iff_components_subset_singleton:
   "connected_space X ⟷ (∃a. connected_components_of X ⊆ {a})"
  by (simp add: connected_components_of_subset_sing)

lemma connected_components_of_eq_singleton:
   "connected_components_of X = {S} ⟷ connected_space X ∧ X ≠ trivial_topology ∧ S = topspace X"
  by (metis connected_components_of_eq_empty connected_components_of_subset_sing insert_not_empty subset_singleton_iff)

lemma connected_components_of_connected_space:
   "connected_space X ⟹ connected_components_of X = (if X = trivial_topology then {} else {topspace X})"
  by (simp add: connected_components_of_eq_empty connected_components_of_eq_singleton)

lemma exists_connected_component_of_superset:
  assumes "connectedin X S" and ne: "X ≠ trivial_topology"
  shows "∃C. C ∈ connected_components_of X ∧ S ⊆ C"
proof (cases "S = {}")
  case True
  then show ?thesis
    using ne connected_components_of_eq_empty by fastforce
next
  case False
  then show ?thesis
    by (meson all_not_in_conv assms(1) connected_component_in_connected_components_of connected_component_of_maximal connectedin_subset_topspace in_mono)
qed

lemma closedin_connected_components_of:
  assumes "C ∈ connected_components_of X"
  shows   "closedin X C"
proof -
  obtain x where "x ∈ topspace X" and x: "C = connected_component_of_set X x"
    using assms by (auto simp: connected_components_of_def)
  have "connected_component_of_set X x ⊆ topspace X"
    by (simp add: connected_component_of_subset_topspace)
  moreover have "X closure_of connected_component_of_set X x ⊆ connected_component_of_set X x"
  proof (rule connected_component_of_maximal)
    show "connectedin X (X closure_of connected_component_of_set X x)"
      by (simp add: connectedin_closure_of connectedin_connected_component_of)
    show "x ∈ X closure_of connected_component_of_set X x"
      by (simp add: ‹x ∈ topspace X› closure_of connected_component_of_refl)
  qed
  ultimately
  show ?thesis
    using closure_of_subset_eq x by auto
qed

lemma closedin_connected_component_of:
   "closedin X (connected_component_of_set X x)"
  by (metis closedin_connected_components_of closedin_empty connected_component_in_connected_components_of connected_component_of_eq_empty)

lemma connected_component_of_eq_overlap:
   "connected_component_of_set X x = connected_component_of_set X y ⟷
      (x ∉ topspace X) ∧ (y ∉ topspace X) ∨
      ~(connected_component_of_set X x ∩ connected_component_of_set X y = {})"
  using connected_component_of_equiv by fastforce

lemma connected_component_of_nonoverlap:
   "connected_component_of_set X x ∩ connected_component_of_set X y = {} ⟷
     (x ∉ topspace X) ∨ (y ∉ topspace X) ∨
     ~(connected_component_of_set X x = connected_component_of_set X y)"
  by (metis connected_component_of_eq_empty connected_component_of_eq_overlap inf.idem)

lemma connected_component_of_overlap:
   "~(connected_component_of_set X x ∩ connected_component_of_set X y = {}) ⟷
    x ∈ topspace X ∧ y ∈ topspace X ∧
    connected_component_of_set X x = connected_component_of_set X y"
  by (meson connected_component_of_nonoverlap)

subsection‹Combining theorems for continuous functions into the reals›

text ‹The homeomorphism between the real line and the open interval $(-1,1)$›

lemma continuous_map_real_shrink:
  "continuous_map euclideanreal (top_of_set {-1<..<1}) (λx. x / (1 + ¦x¦))"
proof -
  have "continuous_on UNIV (λx::real. x / (1 + ¦x¦))"
    by (intro continuous_intros) auto
  then show ?thesis
    by (auto simp: continuous_map_in_subtopology divide_simps)
qed

lemma continuous_on_real_grow:
  "continuous_on {-1<..<1} (λx::real. x / (1 - ¦x¦))"
  by (intro continuous_intros) auto

lemma real_grow_shrink:
  fixes x::real 
  shows "x / (1 + ¦x¦) / (1 - ¦x / (1 + ¦x¦)¦) = x"
  by (force simp: divide_simps)

lemma homeomorphic_maps_real_shrink:
  "homeomorphic_maps euclideanreal (subtopology euclideanreal {-1<..<1}) 
     (λx. x / (1 + ¦x¦))  (λy. y / (1 - ¦y¦))"
  by (force simp: homeomorphic_maps_def continuous_map_real_shrink continuous_on_real_grow divide_simps)

lemma real_shrink_Galois:
  fixes x::real
  shows "(x / (1 + ¦x¦) = y) ⟷ (¦y¦ < 1 ∧ y / (1 - ¦y¦) = x)"
  using real_grow_shrink by (fastforce simp add: distrib_left)

lemma real_shrink_eq:
  fixes x y::real
  shows "(x / (1 + ¦x¦) = y / (1 + ¦y¦)) ⟷ x = y"
  by (metis real_shrink_Galois)

lemma real_shrink_lt:
  fixes x::real
  shows "x / (1 + ¦x¦) < y / (1 + ¦y¦) ⟷ x < y"
  using zero_less_mult_iff [of x y] by (auto simp: field_simps abs_if not_less)

lemma real_shrink_le:
  fixes x::real
  shows "x / (1 + ¦x¦) ≤ y / (1 + ¦y¦) ⟷ x ≤ y"
  by (meson linorder_not_le real_shrink_lt)

lemma real_shrink_grow:
  fixes x::real
  shows "¦x¦ < 1 ⟹ x / (1 - ¦x¦) / (1 + ¦x / (1 - ¦x¦)¦) = x"
  using real_shrink_Galois by blast

lemma continuous_shrink:
  "continuous_on UNIV (λx::real. x / (1 + ¦x¦))"
  by (intro continuous_intros) auto

lemma strict_mono_shrink:
  "strict_mono (λx::real. x / (1 + ¦x¦))"
  by (simp add: monotoneI real_shrink_lt)

lemma shrink_range: "(λx::real. x / (1 + ¦x¦)) ` S ⊆ {-1<..<1}"
  by (auto simp: divide_simps)

text ‹Note: connected sets of real numbers are the same thing as intervals›
lemma connected_shrink:
  fixes S :: "real set"
  shows "connected ((λx. x / (1 + ¦x¦)) ` S) ⟷ connected S"  (is "?lhs = ?rhs")
proof 
  assume "?lhs"
  then have "connected ((λx. x / (1 - ¦x¦)) ` (λx. x / (1 + ¦x¦)) ` S)"
    by (metis continuous_on_real_grow shrink_range connected_continuous_image 
               continuous_on_subset)
  then show "?rhs"
    using real_grow_shrink by (force simp add: image_comp)
next
  assume ?rhs
  then show ?lhs
    using connected_continuous_image 
    by (metis continuous_on_subset continuous_shrink subset_UNIV)
qed

subsection ‹A few cardinality results›

lemma eqpoll_real_subset:
  fixes a b::real
  assumes "a < b" and S: "⋀x. ⟦a < x; x < b⟧ ⟹ x ∈ S"
  shows "S ≈ (UNIV::real set)"
proof (rule lepoll_antisym)
  show "S ≲ (UNIV::real set)"
    by (simp add: subset_imp_lepoll)
  define f where "f ≡ λx. (a+b) / 2 + (b-a) / 2 * (x / (1 + ¦x¦))"
  show "(UNIV::real set) ≲ S"
    unfolding lepoll_def
  proof (intro exI conjI)
    show "inj f"
      unfolding inj_on_def f_def
      by (smt (verit, ccfv_SIG) real_shrink_eq ‹a<b› divide_eq_0_iff mult_cancel_left times_divide_eq_right)
    have pos: "(b-a) / 2 > 0"
      using ‹a<b› by auto
    have *: "a < (a + b) / 2 + (b - a) / 2 * x ⟷ (b - a) > (b - a) * -x"
            "(a + b) / 2 + (b - a) / 2 * x < b ⟷ (b - a) * x < (b - a)" for x
      by (auto simp: field_simps)
    show "range f ⊆ S"
      using shrink_range [of UNIV] ‹a < b›
      unfolding subset_iff f_def greaterThanLessThan_iff image_iff
      by (smt (verit, best) S * mult_less_cancel_left2 mult_minus_right)
  qed
qed

lemma reals01_lepoll_nat_sets: "{0..<1::real} ≲ (UNIV::nat set set)"
proof -
  define nxt where "nxt ≡ λx::real. if x < 1/2 then (True, 2*x) else (False, 2*x - 1)"
  have nxt_fun: "nxt ∈ {0..<1} → UNIV × {0..<1}"
    by (simp add: nxt_def Pi_iff)
  define σ where "σ ≡ λx. rec_nat (True, x) (λn (b,y). nxt y)"
  have σSuc [simp]: "σ x (Suc k) = nxt (snd (σ x k))" for k x
    by (simp add: σ_def case_prod_beta')
  have σ01: "x ∈ {0..<1} ⟹ σ x n ∈ UNIV × {0..<1}" for x n
  proof (induction n)
    case 0
    then show ?case                                           
      by (simp add: σ_def)
   next
    case (Suc n)
    with nxt_fun show ?case
      by (force simp add: Pi_iff split: prod.split)
  qed
  define f where "f ≡ λx. {n. fst (σ x (Suc n))}"
  have snd_nxt: "snd (nxt y) - snd (nxt x) = 2 * (y-x)" 
    if "fst (nxt x) = fst (nxt y)" for x y
    using that by (simp add: nxt_def split: if_split_asm)
  have False if "f x = f y" "x < y" "0 ≤ x" "x < 1" "0 ≤ y" "y < 1" for x y :: real
  proof -
    have "⋀k. fst (σ x (Suc k)) = fst (σ y (Suc k))"
      using that by (force simp add: f_def)
    then have eq: "⋀k. fst (nxt (snd (σ x k))) = fst (nxt (snd (σ y k)))"
      by (metis σ_def case_prod_beta' rec_nat_Suc_imp)
    have *: "snd (σ y k) - snd (σ x k) = 2 ^ k * (y-x)" for k
    proof (induction k)
      case 0
      then show ?case
        by (simp add: σ_def)
    next
      case (Suc k)
      then show ?case
        by (simp add: eq snd_nxt)
    qed
    define n where "n ≡ nat (⌈log 2 (1 / (y - x))⌉)"
    have "2^n ≥ 1 / (y - x)"
      by (simp add: n_def power_of_nat_log_ge)
    then have "2^n * (y-x) ≥ 1"
      using ‹x < y› by (simp add: n_def field_simps)
    with * have "snd (σ y n) - snd (σ x n) ≥ 1"
      by presburger
    moreover have "snd (σ x n) ∈ {0..<1}" "snd (σ y n) ∈ {0..<1}"
      using that by (meson σ01 atLeastLessThan_iff mem_Times_iff)+
    ultimately show False by simp
  qed
  then have "inj_on f {0..<1}"
    by (meson atLeastLessThan_iff linorder_inj_onI')
  then show ?thesis
    unfolding lepoll_def by blast
qed

lemma nat_sets_lepoll_reals01: "(UNIV::nat set set) ≲ {0..<1::real}"
proof -
  define F where "F ≡ λS i. if i∈S then (inverse 3::real) ^ i else 0"
  have Fge0: "F S i ≥ 0" for S i
    by (simp add: F_def)
  have F: "summable (F S)" for S
    unfolding F_def by (force intro: summable_comparison_test_ev [where g = "power (inverse 3)"])
  have "sum (F S) {..<n} ≤ 3/2" for n S
  proof (cases n)
    case (Suc n')
    have "sum (F S) {..<n} ≤ (∑i<n. inverse 3 ^ i)"
      by (simp add: F_def sum_mono)
    also have "… = (∑i=0..n'. inverse 3 ^ i)"
      using Suc atLeast0AtMost lessThan_Suc_atMost by presburger
    also have "… = (3/2) * (1 - inverse 3 ^ n)"
      using sum_gp_multiplied [of 0 n' "inverse (3::real)"] by (simp add: Suc field_simps)
    also have "… ≤ 3/2"
      by (simp add: field_simps)
    finally show ?thesis .
  qed auto
  then have F32: "suminf (F S) ≤ 3/2" for S
    using F suminf_le_const by blast
  define f where "f ≡ λS. suminf (F S) / 2"
  have monoF: "F S n ≤ F T n" if "S ⊆ T" for S T n
    using F_def that by auto
  then have monof: "f S ≤ f T" if "S ⊆ T" for S T
    using that F by (simp add: f_def suminf_le)
  have "f S ∈ {0..<1::real}" for S
  proof -
    have "0 ≤ suminf (F S)"
      using F by (simp add: F_def suminf_nonneg)
    with F32[of S] show ?thesis
      by (auto simp: f_def)
  qed
  moreover have "inj f"
  proof
    fix S T
    assume "f S = f T" 
    show "S = T"
    proof (rule ccontr)
      assume "S ≠ T"
      then have ST_ne: "(S-T) ∪ (T-S) ≠ {}"
        by blast
      define n where "n ≡ LEAST n. n ∈ (S-T) ∪ (T-S)"
      have sum_split: "suminf (F U) = sum (F U) {..<Suc n} + (∑k. F U (k + Suc n))"  for U
        by (metis F add.commute suminf_split_initial_segment)
      have yes: "f U ≥ (sum (F U) {..<n} + (inverse 3::real) ^ n) / 2" 
        if "n ∈ U" for U
      proof -
        have "0 ≤ (∑k. F U (k + Suc n))"
          by (metis F Fge0 suminf_nonneg summable_iff_shift)
        moreover have "F U n = (1/3) ^ n"
          by (simp add: F_def that)
        ultimately show ?thesis
          by (simp add: sum_split f_def)
      qed
      have *: "(∑k. F UNIV (k + n)) = (∑k. F UNIV k) * (inverse 3::real) ^ n" for n
        by (simp add: F_def power_add suminf_mult2)
      have no: "f U < (sum (F U) {..<n} + (inverse 3::real) ^ n) / 2" 
        if "n ∉ U" for U
      proof -
        have [simp]: "F U n = 0"
          by (simp add: F_def that)
        have "(∑k. F U (k + Suc n)) ≤ (∑k. F UNIV (k + Suc n))"
          by (metis F monoF subset_UNIV suminf_le summable_ignore_initial_segment)
        then have "suminf (F U) ≤ (∑k. F UNIV (k + Suc n)) + (∑i<n. F U i)"
          by (simp add: sum_split)
        also have "… < (inverse 3::real) ^ n + (∑i<n. F U i)"
          unfolding * using F32[of UNIV] by simp
        finally have "suminf (F U) < inverse 3 ^ n + sum (F U) {..<n}" .
        then show ?thesis
          by (simp add: f_def)
      qed
      have "S ∩ {..<n} = T ∩ {..<n}"
        using not_less_Least by (fastforce simp add: n_def)
      then have "sum (F S) {..<n} = sum (F T) {..<n}"
        by (metis (no_types, lifting) F_def Int_iff sum.cong)
      moreover consider "n ∈ S-T" | "n ∈ T-S"
        by (metis LeastI_ex ST_ne UnE ex_in_conv n_def)
      ultimately show False
        by (smt (verit, best) Diff_iff ‹f S = f T› yes no)
    qed
  qed
  ultimately show ?thesis
    by (meson image_subsetI lepoll_def)
qed

lemma open_interval_eqpoll_reals:
  fixes a b::real
  shows "{a<..<b} ≈ (UNIV::real set) ⟷ a<b"
  using bij_betw_tan bij_betw_open_intervals eqpoll_def
  by (smt (verit, best) UNIV_I eqpoll_real_subset eqpoll_iff_bijections greaterThanLessThan_iff)

lemma closed_interval_eqpoll_reals:
  fixes a b::real
  shows "{a..b} ≈ (UNIV::real set) ⟷ a < b"
proof
  show "{a..b} ≈ (UNIV::real set) ⟹ a < b"
    using eqpoll_finite_iff infinite_Icc_iff infinite_UNIV_char_0 by blast
qed (auto simp: eqpoll_real_subset)


lemma reals_lepoll_reals01: "(UNIV::real set) ≲ {0..<1::real}"
proof -
  have "(UNIV::real set) ≈ {0<..<1::real}"
    by (simp add: open_interval_eqpoll_reals eqpoll_sym)
  also have "… ≲ {0..<1::real}"
    by (simp add: greaterThanLessThan_subseteq_atLeastLessThan_iff subset_imp_lepoll)
  finally show ?thesis .
qed

lemma nat_sets_eqpoll_reals: "(UNIV::nat set set) ≈ (UNIV::real set)"
  by (metis (mono_tags, opaque_lifting) reals_lepoll_reals01 lepoll_antisym lepoll_trans 
      nat_sets_lepoll_reals01 reals01_lepoll_nat_sets subset_UNIV subset_imp_lepoll)

end