Theory 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: "x2  y2 + z2"
  shows "x  y + z"
proof -
  have "y2 + z2  y2 + 2 * y * z + z2"
    using z y by simp
  with xy have th: "x2  (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  (yU. 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" "yU. indicator A y  ({0<..<2} :: real set)"
      using open_greaterThanLessThan by metis
    hence "yU. 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" "yU. indicator A y  ({-1<..<1} :: real set)"
      using 1 open_greaterThanLessThan by metis
    hence "yU. 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 "yU. 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. (cC. openin (top_of_set S) c)  S  C 
      (DC. finite D  S  D))"
proof safe
  fix C
  assume "compact S" and "cC. 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 "DC. finite D  S  D" ..
next
  assume 1: "C. (cC. openin (top_of_set S) c)  S  C 
        (DC. finite D  S  D)"
  show "compact S"
  proof (rule compactI)
    fix C
    let ?C = "image (λT. S  T) C"
    assume "tC. 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 "DC. 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

subsectiontag 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)

subsectiontag unimportant› ‹A function constant on a set›

definition constant_on  (infixl "(constant'_on)" 50)
  where "f constant_on A  y. xA. 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. xS  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


subsectiontag 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])


subsectiontag 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

subsectiontag unimportant› ‹Seperability›

lemma subset_second_countable:
  obtains  :: "'a:: second_countable_topology set set"
    where "countable "
          "{}  "
          "C. C    openin(top_of_set S) C"
          "T. openin(top_of_set S) T  𝒰. 𝒰    T = 𝒰"
proof -
  obtain  :: "'a set set"
    where "countable "
      and opeB: "C. C    openin(top_of_set S) C"
      and :    "T. openin(top_of_set S) T  𝒰. 𝒰    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 ( - {{}})"
      using countable  by blast
    show "C. C   - {{}}  openin (top_of_set S) C"
      by (simp add: C. C    openin (top_of_set S) C)
    show "𝒰 - {{}}. T = 𝒰" if "openin (top_of_set S) T" for T
      using  [OF that]
      apply clarify
      by (rule_tac x="𝒰 - {{}}" in exI, auto)
  qed auto
qed

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


subsectiontag 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

subsectiontag 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


subsectiontag 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)

subsectiontag 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: "(iI. 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›

definitiontag 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  (xT. r x = x)"

definitiontag 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  xS. f x = x"
    and contg: "continuous_on T g"
    and "g  T  T"
  obtains y where "y  T" and "g y = y"
proof -
  have "xS. (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  (xS. f x = x)) 
         (g. continuous_on T g  g  T  T  (yT. g y = y))"
         (is "?lhs = ?rhs")
proof -
  obtain r i where r:
      "xS. i (r x) = x" "r ` S = T" "continuous_on S r"
      "yT. 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  xS. 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 = {xT. 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