src/HOL/Analysis/Abstract_Topology.thy

tuned proofs

(*  Author:     L C Paulson, University of Cambridge [ported from HOL Light]
*)

section \<open>Operators involving abstract topology\<close>

theory Abstract_Topology
  imports
    Complex_Main
    "HOL-Library.Set_Idioms"
    "HOL-Library.FuncSet"
    (* Path_Connected *)
begin

subsection \<open>General notion of a topology as a value\<close>

definition%important istopology :: "('a set \<Rightarrow> bool) \<Rightarrow> bool" where
"istopology L \<longleftrightarrow>
  L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"

typedef%important 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
  morphisms "openin" "topology"
  unfolding istopology_def by blast

lemma istopology_openin[intro]: "istopology(openin U)"
  using openin[of U] by blast

lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
  using topology_inverse[unfolded mem_Collect_eq] .

lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
  using topology_inverse[of U] istopology_openin[of "topology U"] by auto

lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
proof
  assume "T1 = T2"
  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
next
  assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
  then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
  then have "topology (openin T1) = topology (openin T2)" by simp
  then show "T1 = T2" unfolding openin_inverse .
qed


text\<open>The "universe": the union of all sets in the topology.\<close>
definition "topspace T = \<Union>{S. openin T S}"

subsubsection \<open>Main properties of open sets\<close>

proposition openin_clauses:
  fixes U :: "'a topology"
  shows
    "openin U {}"
    "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
    "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
  using openin[of U] unfolding istopology_def mem_Collect_eq by fast+

lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
  unfolding topspace_def by blast

lemma openin_empty[simp]: "openin U {}"
  by (rule openin_clauses)

lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
  by (rule openin_clauses)

lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
  using openin_clauses by blast

lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
  using openin_Union[of "{S,T}" U] by auto

lemma openin_topspace[intro, simp]: "openin U (topspace U)"
  by (force simp: openin_Union topspace_def)

lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
  (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume ?lhs
  then show ?rhs by auto
next
  assume H: ?rhs
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
  have "openin U ?t" by (force simp: openin_Union)
  also have "?t = S" using H by auto
  finally show "openin U S" .
qed

lemma openin_INT [intro]:
  assumes "finite I"
          "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
  shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)

lemma openin_INT2 [intro]:
  assumes "finite I" "I \<noteq> {}"
          "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
  shows "openin T (\<Inter>i \<in> I. U i)"
proof -
  have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
    using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
  then show ?thesis
    using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
qed

lemma openin_Inter [intro]:
  assumes "finite \<F>" "\<F> \<noteq> {}" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter>\<F>)"
  by (metis (full_types) assms openin_INT2 image_ident)

lemma openin_Int_Inter:
  assumes "finite \<F>" "openin T U" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter>\<F>)"
  using openin_Inter [of "insert U \<F>"] assms by auto


subsubsection \<open>Closed sets\<close>

definition%important closedin :: "'a topology \<Rightarrow> 'a set \<Rightarrow> bool" where
"closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"

lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
  by (metis closedin_def)

lemma closedin_empty[simp]: "closedin U {}"
  by (simp add: closedin_def)

lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
  by (simp add: closedin_def)

lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
  by (auto simp: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
  by auto

lemma closedin_Union:
  assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
    shows "closedin U (\<Union>S)"
  using assms by induction auto

lemma closedin_Inter[intro]:
  assumes Ke: "K \<noteq> {}"
    and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
  shows "closedin U (\<Inter>K)"
  using Ke Kc unfolding closedin_def Diff_Inter by auto

lemma closedin_INT[intro]:
  assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
  shows "closedin U (\<Inter>x\<in>A. B x)"
  apply (rule closedin_Inter)
  using assms
  apply auto
  done

lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
  using closedin_Inter[of "{S,T}" U] by auto

lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
  apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
  apply (metis openin_subset subset_eq)
  done

lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
  by (simp add: openin_closedin_eq)

lemma openin_diff[intro]:
  assumes oS: "openin U S"
    and cT: "closedin U T"
  shows "openin U (S - T)"
proof -
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
    by (auto simp: topspace_def openin_subset)
  then show ?thesis using oS cT
    by (auto simp: closedin_def)
qed

lemma closedin_diff[intro]:
  assumes oS: "closedin U S"
    and cT: "openin U T"
  shows "closedin U (S - T)"
proof -
  have "S - T = S \<inter> (topspace U - T)"
    using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
  then show ?thesis
    using oS cT by (auto simp: openin_closedin_eq)
qed


subsection\<open>The discrete topology\<close>

definition discrete_topology where "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"

lemma openin_discrete_topology [simp]: "openin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
proof -
  have "istopology (\<lambda>S. S \<subseteq> U)"
    by (auto simp: istopology_def)
  then show ?thesis
    by (simp add: discrete_topology_def topology_inverse')
qed

lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U"
  by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym)

lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
  by (simp add: closedin_def)

lemma discrete_topology_unique:
   "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs")
proof
  assume R: ?rhs
  then have "openin X S" if "S \<subseteq> U" for S
    using openin_subopen subsetD that by fastforce
  moreover have "x \<in> topspace X" if "openin X S" and "x \<in> S" for x S
    using openin_subset that by blast
  ultimately
  show ?lhs
    using R by (auto simp: topology_eq)
qed auto

lemma discrete_topology_unique_alt:
  "discrete_topology U = X \<longleftrightarrow> topspace X \<subseteq> U \<and> (\<forall>x \<in> U. openin X {x})"
  using openin_subset
  by (auto simp: discrete_topology_unique)

lemma subtopology_eq_discrete_topology_empty:
   "X = discrete_topology {} \<longleftrightarrow> topspace X = {}"
  using discrete_topology_unique [of "{}" X] by auto

lemma subtopology_eq_discrete_topology_sing:
   "X = discrete_topology {a} \<longleftrightarrow> topspace X = {a}"
  by (metis discrete_topology_unique openin_topspace singletonD)


subsection \<open>Subspace topology\<close>

definition%important subtopology :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a topology" where
"subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
  (is "istopology ?L")
proof -
  have "?L {}" by blast
  {
    fix A B
    assume A: "?L A" and B: "?L B"
    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V"
      by blast
    have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
      using Sa Sb by blast+
    then have "?L (A \<inter> B)" by blast
  }
  moreover
  {
    fix K
    assume K: "K \<subseteq> Collect ?L"
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
      by blast
    from K[unfolded th0 subset_image_iff]
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
      by blast
    have "\<Union>K = (\<Union>Sk) \<inter> V"
      using Sk by auto
    moreover have "openin U (\<Union>Sk)"
      using Sk by (auto simp: subset_eq)
    ultimately have "?L (\<Union>K)" by blast
  }
  ultimately show ?thesis
    unfolding subset_eq mem_Collect_eq istopology_def by auto
qed

lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
  by auto

lemma openin_subtopology_Int:
   "openin X S \<Longrightarrow> openin (subtopology X T) (S \<inter> T)"
  using openin_subtopology by auto

lemma openin_subtopology_Int2:
   "openin X T \<Longrightarrow> openin (subtopology X S) (S \<inter> T)"
  using openin_subtopology by auto

lemma openin_subtopology_diff_closed:
   "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> openin (subtopology X S) (S - T)"
  unfolding closedin_def openin_subtopology
  by (rule_tac x="topspace X - T" in exI) auto

lemma openin_relative_to: "(openin X relative_to S) = openin (subtopology X S)"
  by (force simp: relative_to_def openin_subtopology)

lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
  by (auto simp: topspace_def openin_subtopology)

lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
  unfolding closedin_def topspace_subtopology
  by (auto simp: openin_subtopology)

lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
  unfolding openin_subtopology
  by auto (metis IntD1 in_mono openin_subset)

lemma subtopology_subtopology:
   "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
proof -
  have eq: "\<And>T'. (\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
    by (metis inf_assoc)
  have "subtopology (subtopology X S) T = topology (\<lambda>Ta. \<exists>Sa. Ta = Sa \<inter> T \<and> openin (subtopology X S) Sa)"
    by (simp add: subtopology_def)
  also have "\<dots> = subtopology X (S \<inter> T)"
    by (simp add: openin_subtopology eq) (simp add: subtopology_def)
  finally show ?thesis .
qed

lemma openin_subtopology_alt:
     "openin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (openin X)"
  by (simp add: image_iff inf_commute openin_subtopology)

lemma closedin_subtopology_alt:
     "closedin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (closedin X)"
  by (simp add: image_iff inf_commute closedin_subtopology)

lemma subtopology_superset:
  assumes UV: "topspace U \<subseteq> V"
  shows "subtopology U V = U"
proof -
  {
    fix S
    {
      fix T
      assume T: "openin U T" "S = T \<inter> V"
      from T openin_subset[OF T(1)] UV have eq: "S = T"
        by blast
      have "openin U S"
        unfolding eq using T by blast
    }
    moreover
    {
      assume S: "openin U S"
      then have "\<exists>T. openin U T \<and> S = T \<inter> V"
        using openin_subset[OF S] UV by auto
    }
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
      by blast
  }
  then show ?thesis
    unfolding topology_eq openin_subtopology by blast
qed

lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
  by (simp add: subtopology_superset)

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
  by (simp add: subtopology_superset)

lemma openin_subtopology_empty:
   "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
by (metis Int_empty_right openin_empty openin_subtopology)

lemma closedin_subtopology_empty:
   "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
by (metis Int_empty_right closedin_empty closedin_subtopology)

lemma closedin_subtopology_refl [simp]:
   "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)

lemma closedin_topspace_empty: "topspace T = {} \<Longrightarrow> (closedin T S \<longleftrightarrow> S = {})"
  by (simp add: closedin_def)

lemma openin_imp_subset:
   "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
by (metis Int_iff openin_subtopology subsetI)

lemma closedin_imp_subset:
   "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
by (simp add: closedin_def topspace_subtopology)

lemma openin_open_subtopology:
     "openin X S \<Longrightarrow> openin (subtopology X S) T \<longleftrightarrow> openin X T \<and> T \<subseteq> S"
  by (metis inf.orderE openin_Int openin_imp_subset openin_subtopology)

lemma closedin_closed_subtopology:
     "closedin X S \<Longrightarrow> (closedin (subtopology X S) T \<longleftrightarrow> closedin X T \<and> T \<subseteq> S)"
  by (metis closedin_Int closedin_imp_subset closedin_subtopology inf.orderE)

lemma openin_subtopology_Un:
    "\<lbrakk>openin (subtopology X T) S; openin (subtopology X U) S\<rbrakk>
     \<Longrightarrow> openin (subtopology X (T \<union> U)) S"
by (simp add: openin_subtopology) blast

lemma closedin_subtopology_Un:
    "\<lbrakk>closedin (subtopology X T) S; closedin (subtopology X U) S\<rbrakk>
     \<Longrightarrow> closedin (subtopology X (T \<union> U)) S"
by (simp add: closedin_subtopology) blast


subsection \<open>The standard Euclidean topology\<close>

definition%important euclidean :: "'a::topological_space topology"
  where "euclidean = topology open"

lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
  unfolding euclidean_def
  apply (rule cong[where x=S and y=S])
  apply (rule topology_inverse[symmetric])
  apply (auto simp: istopology_def)
  done

declare open_openin [symmetric, simp]

lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
  by (force simp: topspace_def)

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
  by (simp add: topspace_subtopology)

lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
  by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)

declare closed_closedin [symmetric, simp]

lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
  by (metis openin_topspace topspace_euclidean_subtopology)

subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>

abbreviation euclideanreal :: "real topology"
  where "euclideanreal \<equiv> topology open"

lemma real_openin [simp]: "openin euclideanreal S = open S"
  by (simp add: euclidean_def open_openin)

lemma topspace_euclideanreal [simp]: "topspace euclideanreal = UNIV"
  using openin_subset open_UNIV real_openin by blast

lemma topspace_euclideanreal_subtopology [simp]:
   "topspace (subtopology euclideanreal S) = S"
  by (simp add: topspace_subtopology)

lemma real_closedin [simp]: "closedin euclideanreal S = closed S"
  by (simp add: closed_closedin euclidean_def)

subsection \<open>Basic "localization" results are handy for connectedness.\<close>

lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
  by (auto simp: openin_subtopology)

lemma openin_Int_open:
   "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
        \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
by (metis open_Int Int_assoc openin_open)

lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
  by (auto simp: openin_open)

lemma open_openin_trans[trans]:
  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
  by (metis Int_absorb1  openin_open_Int)

lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
  by (auto simp: openin_open)

lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
  by (simp add: closedin_subtopology Int_ac)

lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
  by (metis closedin_closed)

lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
  by (auto simp: closedin_closed)

lemma closedin_closed_subset:
 "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
             \<Longrightarrow> closedin (subtopology euclidean T) S"
  by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)

lemma finite_imp_closedin:
  fixes S :: "'a::t1_space set"
  shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
    by (simp add: finite_imp_closed closed_subset)

lemma closedin_singleton [simp]:
  fixes a :: "'a::t1_space"
  shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
using closedin_subset  by (force intro: closed_subset)

lemma openin_euclidean_subtopology_iff:
  fixes S U :: "'a::metric_space set"
  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
    S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
  (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding openin_open open_dist by blast
next
  define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
    unfolding T_def
    apply clarsimp
    apply (rule_tac x="d - dist x a" in exI)
    apply (clarsimp simp add: less_diff_eq)
    by (metis dist_commute dist_triangle_lt)
  assume ?rhs then have 2: "S = U \<inter> T"
    unfolding T_def
    by auto (metis dist_self)
  from 1 2 show ?lhs
    unfolding openin_open open_dist by fast
qed

lemma connected_openin:
      "connected S \<longleftrightarrow>
       \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
                 openin (subtopology euclidean S) E2 \<and>
                 S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
  apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
  by (simp_all, blast+)  (* SLOW *)

lemma connected_openin_eq:
      "connected S \<longleftrightarrow>
       \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
                 openin (subtopology euclidean S) E2 \<and>
                 E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
                 E1 \<noteq> {} \<and> E2 \<noteq> {})"
  apply (simp add: connected_openin, safe, blast)
  by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)

lemma connected_closedin:
      "connected S \<longleftrightarrow>
       (\<nexists>E1 E2.
        closedin (subtopology euclidean S) E1 \<and>
        closedin (subtopology euclidean S) E2 \<and>
        S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
       (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs 
    by (auto simp add: connected_closed closedin_closed)
next
  assume R: ?rhs
  then show ?lhs 
  proof (clarsimp simp add: connected_closed closedin_closed)
    fix A B 
    assume s_sub: "S \<subseteq> A \<union> B" "B \<inter> S \<noteq> {}"
      and disj: "A \<inter> B \<inter> S = {}"
      and cl: "closed A" "closed B"
    have "S \<inter> (A \<union> B) = S"
      using s_sub(1) by auto
    have "S - A = B \<inter> S"
      using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
    then have "S \<inter> A = {}"
      by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
    then show "A \<inter> S = {}"
      by blast
  qed
qed

lemma connected_closedin_eq:
      "connected S \<longleftrightarrow>
           \<not>(\<exists>E1 E2.
                 closedin (subtopology euclidean S) E1 \<and>
                 closedin (subtopology euclidean S) E2 \<and>
                 E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
                 E1 \<noteq> {} \<and> E2 \<noteq> {})"
  apply (simp add: connected_closedin, safe, blast)
  by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)

text \<open>These "transitivity" results are handy too\<close>

lemma openin_trans[trans]:
  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
    openin (subtopology euclidean U) S"
  unfolding open_openin openin_open by blast

lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
  by (auto simp: openin_open intro: openin_trans)

lemma closedin_trans[trans]:
  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
    closedin (subtopology euclidean U) S"
  by (auto simp: closedin_closed closed_Inter Int_assoc)

lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
  by (auto simp: closedin_closed intro: closedin_trans)

lemma openin_subtopology_Int_subset:
   "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
  by (auto simp: openin_subtopology)

lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
  using open_subset openin_open_trans openin_subset by fastforce


subsection\<open>Derived set (set of limit points)\<close>

definition derived_set_of :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "derived'_set'_of" 80)
  where "X derived_set_of S \<equiv>
         {x \<in> topspace X.
                (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<noteq>x. y \<in> S \<and> y \<in> T))}"

lemma derived_set_of_restrict:
   "X derived_set_of (topspace X \<inter> S) = X derived_set_of S"
  by (simp add: derived_set_of_def) (metis openin_subset subset_iff)

lemma in_derived_set_of:
   "x \<in> X derived_set_of S \<longleftrightarrow> x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<noteq>x. y \<in> S \<and> y \<in> T))"
  by (simp add: derived_set_of_def)

lemma derived_set_of_subset_topspace:
   "X derived_set_of S \<subseteq> topspace X"
  by (auto simp add: derived_set_of_def)

lemma derived_set_of_subtopology:
   "(subtopology X U) derived_set_of S = U \<inter> (X derived_set_of (U \<inter> S))"
  by (simp add: derived_set_of_def openin_subtopology topspace_subtopology) blast

lemma derived_set_of_subset_subtopology:
   "(subtopology X S) derived_set_of T \<subseteq> S"
  by (simp add: derived_set_of_subtopology)

lemma derived_set_of_empty [simp]: "X derived_set_of {} = {}"
  by (auto simp: derived_set_of_def)

lemma derived_set_of_mono:
   "S \<subseteq> T \<Longrightarrow> X derived_set_of S \<subseteq> X derived_set_of T"
  unfolding derived_set_of_def by blast

lemma derived_set_of_union:
   "X derived_set_of (S \<union> T) = X derived_set_of S \<union> X derived_set_of T" (is "?lhs = ?rhs")
proof
  show "?lhs \<subseteq> ?rhs"
    apply (clarsimp simp: in_derived_set_of)
    by (metis IntE IntI openin_Int)
  show "?rhs \<subseteq> ?lhs"
    by (simp add: derived_set_of_mono)
qed

lemma derived_set_of_unions:
   "finite \<F> \<Longrightarrow> X derived_set_of (\<Union>\<F>) = (\<Union>S \<in> \<F>. X derived_set_of S)"
proof (induction \<F> rule: finite_induct)
  case (insert S \<F>)
  then show ?case
    by (simp add: derived_set_of_union)
qed auto

lemma derived_set_of_topspace:
  "X derived_set_of (topspace X) = {x \<in> topspace X. \<not> openin X {x}}"
  apply (auto simp: in_derived_set_of)
  by (metis Set.set_insert all_not_in_conv insertCI openin_subset subsetCE)

lemma discrete_topology_unique_derived_set:
     "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> X derived_set_of U = {}"
  by (auto simp: discrete_topology_unique derived_set_of_topspace)

lemma subtopology_eq_discrete_topology_eq:
   "subtopology X U = discrete_topology U \<longleftrightarrow> U \<subseteq> topspace X \<and> U \<inter> X derived_set_of U = {}"
  using discrete_topology_unique_derived_set [of U "subtopology X U"]
  by (auto simp: eq_commute topspace_subtopology derived_set_of_subtopology)

lemma subtopology_eq_discrete_topology:
   "S \<subseteq> topspace X \<and> S \<inter> X derived_set_of S = {}
        \<Longrightarrow> subtopology X S = discrete_topology S"
  by (simp add: subtopology_eq_discrete_topology_eq)

lemma subtopology_eq_discrete_topology_gen:
   "S \<inter> X derived_set_of S = {} \<Longrightarrow> subtopology X S = discrete_topology(topspace X \<inter> S)"
  by (metis Int_lower1 derived_set_of_restrict inf_assoc inf_bot_right subtopology_eq_discrete_topology_eq subtopology_subtopology subtopology_topspace)

lemma openin_Int_derived_set_of_subset:
   "openin X S \<Longrightarrow> S \<inter> X derived_set_of T \<subseteq> X derived_set_of (S \<inter> T)"
  by (auto simp: derived_set_of_def)

lemma openin_Int_derived_set_of_eq:
  "openin X S \<Longrightarrow> S \<inter> X derived_set_of T = S \<inter> X derived_set_of (S \<inter> T)"
  apply auto
   apply (meson IntI openin_Int_derived_set_of_subset subsetCE)
  by (meson derived_set_of_mono inf_sup_ord(2) subset_eq)


subsection\<open> Closure with respect to a topological space\<close>

definition closure_of :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixr "closure'_of" 80)
  where "X closure_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y \<in> S. y \<in> T)}"

lemma closure_of_restrict: "X closure_of S = X closure_of (topspace X \<inter> S)"
  unfolding closure_of_def
  apply safe
  apply (meson IntI openin_subset subset_iff)
  by auto

lemma in_closure_of:
   "x \<in> X closure_of S \<longleftrightarrow>
    x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<in> S \<and> y \<in> T))"
  by (auto simp: closure_of_def)

lemma closure_of: "X closure_of S = topspace X \<inter> (S \<union> X derived_set_of S)"
  by (fastforce simp: in_closure_of in_derived_set_of)

lemma closure_of_alt: "X closure_of S = topspace X \<inter> S \<union> X derived_set_of S"
  using derived_set_of_subset_topspace [of X S]
  unfolding closure_of_def in_derived_set_of
  by safe (auto simp: in_derived_set_of)

lemma derived_set_of_subset_closure_of:
   "X derived_set_of S \<subseteq> X closure_of S"
  by (fastforce simp: closure_of_def in_derived_set_of)

lemma closure_of_subtopology:
  "(subtopology X U) closure_of S = U \<inter> (X closure_of (U \<inter> S))"
  unfolding closure_of_def topspace_subtopology openin_subtopology
  by safe (metis (full_types) IntI Int_iff inf.commute)+

lemma closure_of_empty [simp]: "X closure_of {} = {}"
  by (simp add: closure_of_alt)

lemma closure_of_topspace [simp]: "X closure_of topspace X = topspace X"
  by (simp add: closure_of)

lemma closure_of_UNIV [simp]: "X closure_of UNIV = topspace X"
  by (simp add: closure_of)

lemma closure_of_subset_topspace: "X closure_of S \<subseteq> topspace X"
  by (simp add: closure_of)

lemma closure_of_subset_subtopology: "(subtopology X S) closure_of T \<subseteq> S"
  by (simp add: closure_of_subtopology)

lemma closure_of_mono: "S \<subseteq> T \<Longrightarrow> X closure_of S \<subseteq> X closure_of T"
  by (fastforce simp add: closure_of_def)

lemma closure_of_subtopology_subset:
   "(subtopology X U) closure_of S \<subseteq> (X closure_of S)"
  unfolding closure_of_subtopology
  by clarsimp (meson closure_of_mono contra_subsetD inf.cobounded2)

lemma closure_of_subtopology_mono:
   "T \<subseteq> U \<Longrightarrow> (subtopology X T) closure_of S \<subseteq> (subtopology X U) closure_of S"
  unfolding closure_of_subtopology
  by auto (meson closure_of_mono inf_mono subset_iff)

lemma closure_of_Un [simp]: "X closure_of (S \<union> T) = X closure_of S \<union> X closure_of T"
  by (simp add: Un_assoc Un_left_commute closure_of_alt derived_set_of_union inf_sup_distrib1)

lemma closure_of_Union:
   "finite \<F> \<Longrightarrow> X closure_of (\<Union>\<F>) = (\<Union>S \<in> \<F>. X closure_of S)"
by (induction \<F> rule: finite_induct) auto

lemma closure_of_subset: "S \<subseteq> topspace X \<Longrightarrow> S \<subseteq> X closure_of S"
  by (auto simp: closure_of_def)

lemma closure_of_subset_Int: "topspace X \<inter> S \<subseteq> X closure_of S"
  by (auto simp: closure_of_def)

lemma closure_of_subset_eq: "S \<subseteq> topspace X \<and> X closure_of S \<subseteq> S \<longleftrightarrow> closedin X S"
proof (cases "S \<subseteq> topspace X")
  case True
  then have "\<forall>x. x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<in>S. y \<in> T)) \<longrightarrow> x \<in> S
             \<Longrightarrow> openin X (topspace X - S)"
    apply (subst openin_subopen, safe)
    by (metis DiffI subset_eq openin_subset [of X])
  then show ?thesis
    by (auto simp: closedin_def closure_of_def)
next
  case False
  then show ?thesis
    by (simp add: closedin_def)
qed

lemma closure_of_eq: "X closure_of S = S \<longleftrightarrow> closedin X S"
proof (cases "S \<subseteq> topspace X")
  case True
  then show ?thesis
    by (metis closure_of_subset closure_of_subset_eq set_eq_subset)
next
  case False
  then show ?thesis
    using closure_of closure_of_subset_eq by fastforce
qed

lemma closedin_contains_derived_set:
   "closedin X S \<longleftrightarrow> X derived_set_of S \<subseteq> S \<and> S \<subseteq> topspace X"
proof (intro iffI conjI)
  show "closedin X S \<Longrightarrow> X derived_set_of S \<subseteq> S"
    using closure_of_eq derived_set_of_subset_closure_of by fastforce
  show "closedin X S \<Longrightarrow> S \<subseteq> topspace X"
    using closedin_subset by blast
  show "X derived_set_of S \<subseteq> S \<and> S \<subseteq> topspace X \<Longrightarrow> closedin X S"
    by (metis closure_of closure_of_eq inf.absorb_iff2 sup.orderE)
qed

lemma derived_set_subset_gen:
   "X derived_set_of S \<subseteq> S \<longleftrightarrow> closedin X (topspace X \<inter> S)"
  by (simp add: closedin_contains_derived_set derived_set_of_restrict derived_set_of_subset_topspace)

lemma derived_set_subset: "S \<subseteq> topspace X \<Longrightarrow> (X derived_set_of S \<subseteq> S \<longleftrightarrow> closedin X S)"
  by (simp add: closedin_contains_derived_set)

lemma closedin_derived_set:
     "closedin (subtopology X T) S \<longleftrightarrow>
      S \<subseteq> topspace X \<and> S \<subseteq> T \<and> (\<forall>x. x \<in> X derived_set_of S \<and> x \<in> T \<longrightarrow> x \<in> S)"
  by (auto simp: closedin_contains_derived_set topspace_subtopology derived_set_of_subtopology Int_absorb1)

lemma closedin_Int_closure_of:
     "closedin (subtopology X S) T \<longleftrightarrow> S \<inter> X closure_of T = T"
  by (metis Int_left_absorb closure_of_eq closure_of_subtopology)

lemma closure_of_closedin: "closedin X S \<Longrightarrow> X closure_of S = S"
  by (simp add: closure_of_eq)

lemma closure_of_eq_diff: "X closure_of S = topspace X - \<Union>{T. openin X T \<and> disjnt S T}"
  by (auto simp: closure_of_def disjnt_iff)

lemma closedin_closure_of [simp]: "closedin X (X closure_of S)"
  unfolding closure_of_eq_diff by blast

lemma closure_of_closure_of [simp]: "X closure_of (X closure_of S) = X closure_of S"
  by (simp add: closure_of_eq)

lemma closure_of_hull:
  assumes "S \<subseteq> topspace X" shows "X closure_of S = (closedin X) hull S"
proof (rule hull_unique [THEN sym])
  show "S \<subseteq> X closure_of S"
    by (simp add: closure_of_subset assms)
next
  show "closedin X (X closure_of S)"
    by simp
  show "\<And>T. \<lbrakk>S \<subseteq> T; closedin X T\<rbrakk> \<Longrightarrow> X closure_of S \<subseteq> T"
    by (metis closure_of_eq closure_of_mono)
qed

lemma closure_of_minimal:
   "\<lbrakk>S \<subseteq> T; closedin X T\<rbrakk> \<Longrightarrow> (X closure_of S) \<subseteq> T"
  by (metis closure_of_eq closure_of_mono)

lemma closure_of_minimal_eq:
   "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> (X closure_of S) \<subseteq> T \<longleftrightarrow> S \<subseteq> T"
  by (meson closure_of_minimal closure_of_subset subset_trans)

lemma closure_of_unique:
   "\<lbrakk>S \<subseteq> T; closedin X T;
     \<And>T'. \<lbrakk>S \<subseteq> T'; closedin X T'\<rbrakk> \<Longrightarrow> T \<subseteq> T'\<rbrakk>
    \<Longrightarrow> X closure_of S = T"
  by (meson closedin_closure_of closedin_subset closure_of_minimal closure_of_subset eq_iff order.trans)

lemma closure_of_eq_empty_gen: "X closure_of S = {} \<longleftrightarrow> disjnt (topspace X) S"
  unfolding disjnt_def closure_of_restrict [where S=S]
  using closure_of by fastforce

lemma closure_of_eq_empty: "S \<subseteq> topspace X \<Longrightarrow> X closure_of S = {} \<longleftrightarrow> S = {}"
  using closure_of_subset by fastforce

lemma openin_Int_closure_of_subset:
  assumes "openin X S"
  shows "S \<inter> X closure_of T \<subseteq> X closure_of (S \<inter> T)"
proof -
  have "S \<inter> X derived_set_of T = S \<inter> X derived_set_of (S \<inter> T)"
    by (meson assms openin_Int_derived_set_of_eq)
  moreover have "S \<inter> (S \<inter> T) = S \<inter> T"
    by fastforce
  ultimately show ?thesis
    by (metis closure_of_alt inf.cobounded2 inf_left_commute inf_sup_distrib1)
qed

lemma closure_of_openin_Int_closure_of:
  assumes "openin X S"
  shows "X closure_of (S \<inter> X closure_of T) = X closure_of (S \<inter> T)"
proof
  show "X closure_of (S \<inter> X closure_of T) \<subseteq> X closure_of (S \<inter> T)"
    by (simp add: assms closure_of_minimal openin_Int_closure_of_subset)
next
  show "X closure_of (S \<inter> T) \<subseteq> X closure_of (S \<inter> X closure_of T)"
    by (metis Int_lower1 Int_subset_iff assms closedin_closure_of closure_of_minimal_eq closure_of_mono inf_le2 le_infI1 openin_subset)
qed

lemma openin_Int_closure_of_eq:
  "openin X S \<Longrightarrow> S \<inter> X closure_of T = S \<inter> X closure_of (S \<inter> T)"
  apply (rule equalityI)
   apply (simp add: openin_Int_closure_of_subset)
  by (meson closure_of_mono inf.cobounded2 inf_mono subset_refl)

lemma openin_Int_closure_of_eq_empty:
   "openin X S \<Longrightarrow> S \<inter> X closure_of T = {} \<longleftrightarrow> S \<inter> T = {}"
  apply (subst openin_Int_closure_of_eq, auto)
  by (meson IntI closure_of_subset_Int disjoint_iff_not_equal openin_subset subset_eq)

lemma closure_of_openin_Int_superset:
   "openin X S \<and> S \<subseteq> X closure_of T
        \<Longrightarrow> X closure_of (S \<inter> T) = X closure_of S"
  by (metis closure_of_openin_Int_closure_of inf.orderE)

lemma closure_of_openin_subtopology_Int_closure_of:
  assumes S: "openin (subtopology X U) S" and "T \<subseteq> U"
  shows "X closure_of (S \<inter> X closure_of T) = X closure_of (S \<inter> T)" (is "?lhs = ?rhs")
proof
  obtain S0 where S0: "openin X S0" "S = S0 \<inter> U"
    using assms by (auto simp: openin_subtopology)
  show "?lhs \<subseteq> ?rhs"
  proof -
    have "S0 \<inter> X closure_of T = S0 \<inter> X closure_of (S0 \<inter> T)"
      by (meson S0(1) openin_Int_closure_of_eq)
    moreover have "S0 \<inter> T = S0 \<inter> U \<inter> T"
      using \<open>T \<subseteq> U\<close> by fastforce
    ultimately have "S \<inter> X closure_of T \<subseteq> X closure_of (S \<inter> T)"
      using S0(2) by auto
    then show ?thesis
      by (meson closedin_closure_of closure_of_minimal)
  qed
next
  show "?rhs \<subseteq> ?lhs"
  proof -
    have "T \<inter> S \<subseteq> T \<union> X derived_set_of T"
      by force
    then show ?thesis
      by (metis Int_subset_iff S closure_of closure_of_mono inf.cobounded2 inf.coboundedI2 inf_commute openin_closedin_eq topspace_subtopology)
  qed
qed

lemma closure_of_subtopology_open:
     "openin X U \<or> S \<subseteq> U \<Longrightarrow> (subtopology X U) closure_of S = U \<inter> X closure_of S"
  by (metis closure_of_subtopology inf_absorb2 openin_Int_closure_of_eq)

lemma discrete_topology_closure_of:
     "(discrete_topology U) closure_of S = U \<inter> S"
  by (metis closedin_discrete_topology closure_of_restrict closure_of_unique discrete_topology_unique inf_sup_ord(1) order_refl)


text\<open> Interior with respect to a topological space.                             \<close>

definition interior_of :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixr "interior'_of" 80)
  where "X interior_of S \<equiv> {x. \<exists>T. openin X T \<and> x \<in> T \<and> T \<subseteq> S}"

lemma interior_of_restrict:
   "X interior_of S = X interior_of (topspace X \<inter> S)"
  using openin_subset by (auto simp: interior_of_def)

lemma interior_of_eq: "(X interior_of S = S) \<longleftrightarrow> openin X S"
  unfolding interior_of_def  using openin_subopen by blast

lemma interior_of_openin: "openin X S \<Longrightarrow> X interior_of S = S"
  by (simp add: interior_of_eq)

lemma interior_of_empty [simp]: "X interior_of {} = {}"
  by (simp add: interior_of_eq)

lemma interior_of_topspace [simp]: "X interior_of (topspace X) = topspace X"
  by (simp add: interior_of_eq)

lemma openin_interior_of [simp]: "openin X (X interior_of S)"
  unfolding interior_of_def
  using openin_subopen by fastforce

lemma interior_of_interior_of [simp]:
   "X interior_of X interior_of S = X interior_of S"
  by (simp add: interior_of_eq)

lemma interior_of_subset: "X interior_of S \<subseteq> S"
  by (auto simp: interior_of_def)

lemma interior_of_subset_closure_of: "X interior_of S \<subseteq> X closure_of S"
  by (metis closure_of_subset_Int dual_order.trans interior_of_restrict interior_of_subset)

lemma subset_interior_of_eq: "S \<subseteq> X interior_of S \<longleftrightarrow> openin X S"
  by (metis interior_of_eq interior_of_subset subset_antisym)

lemma interior_of_mono: "S \<subseteq> T \<Longrightarrow> X interior_of S \<subseteq> X interior_of T"
  by (auto simp: interior_of_def)

lemma interior_of_maximal: "\<lbrakk>T \<subseteq> S; openin X T\<rbrakk> \<Longrightarrow> T \<subseteq> X interior_of S"
  by (auto simp: interior_of_def)

lemma interior_of_maximal_eq: "openin X T \<Longrightarrow> T \<subseteq> X interior_of S \<longleftrightarrow> T \<subseteq> S"
  by (meson interior_of_maximal interior_of_subset order_trans)

lemma interior_of_unique:
   "\<lbrakk>T \<subseteq> S; openin X T; \<And>T'. \<lbrakk>T' \<subseteq> S; openin X T'\<rbrakk> \<Longrightarrow> T' \<subseteq> T\<rbrakk> \<Longrightarrow> X interior_of S = T"
  by (simp add: interior_of_maximal_eq interior_of_subset subset_antisym)

lemma interior_of_subset_topspace: "X interior_of S \<subseteq> topspace X"
  by (simp add: openin_subset)

lemma interior_of_subset_subtopology: "(subtopology X S) interior_of T \<subseteq> S"
  by (meson openin_imp_subset openin_interior_of)

lemma interior_of_Int: "X interior_of (S \<inter> T) = X interior_of S \<inter> X interior_of T"
  apply (rule equalityI)
   apply (simp add: interior_of_mono)
  apply (auto simp: interior_of_maximal_eq openin_Int interior_of_subset le_infI1 le_infI2)
  done

lemma interior_of_Inter_subset: "X interior_of (\<Inter>\<F>) \<subseteq> (\<Inter>S \<in> \<F>. X interior_of S)"
  by (simp add: INT_greatest Inf_lower interior_of_mono)

lemma union_interior_of_subset:
   "X interior_of S \<union> X interior_of T \<subseteq> X interior_of (S \<union> T)"
  by (simp add: interior_of_mono)

lemma interior_of_eq_empty:
   "X interior_of S = {} \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<subseteq> S \<longrightarrow> T = {})"
  by (metis bot.extremum_uniqueI interior_of_maximal interior_of_subset openin_interior_of)

lemma interior_of_eq_empty_alt:
   "X interior_of S = {} \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<noteq> {} \<longrightarrow> T - S \<noteq> {})"
  by (auto simp: interior_of_eq_empty)

lemma interior_of_Union_openin_subsets:
   "\<Union>{T. openin X T \<and> T \<subseteq> S} = X interior_of S"
  by (rule interior_of_unique [symmetric]) auto

lemma interior_of_complement:
   "X interior_of (topspace X - S) = topspace X - X closure_of S"
  by (auto simp: interior_of_def closure_of_def)

lemma interior_of_closure_of:
   "X interior_of S = topspace X - X closure_of (topspace X - S)"
  unfolding interior_of_complement [symmetric]
  by (metis Diff_Diff_Int interior_of_restrict)

lemma closure_of_interior_of:
   "X closure_of S = topspace X - X interior_of (topspace X - S)"
  by (simp add: interior_of_complement Diff_Diff_Int closure_of)

lemma closure_of_complement: "X closure_of (topspace X - S) = topspace X - X interior_of S"
  unfolding interior_of_def closure_of_def
  by (blast dest: openin_subset)

lemma interior_of_eq_empty_complement:
  "X interior_of S = {} \<longleftrightarrow> X closure_of (topspace X - S) = topspace X"
  using interior_of_subset_topspace [of X S] closure_of_complement by fastforce

lemma closure_of_eq_topspace:
   "X closure_of S = topspace X \<longleftrightarrow> X interior_of (topspace X - S) = {}"
  using closure_of_subset_topspace [of X S] interior_of_complement by fastforce

lemma interior_of_subtopology_subset:
     "U \<inter> X interior_of S \<subseteq> (subtopology X U) interior_of S"
  by (auto simp: interior_of_def openin_subtopology)

lemma interior_of_subtopology_subsets:
   "T \<subseteq> U \<Longrightarrow> T \<inter> (subtopology X U) interior_of S \<subseteq> (subtopology X T) interior_of S"
  by (metis inf.absorb_iff2 interior_of_subtopology_subset subtopology_subtopology)

lemma interior_of_subtopology_mono:
   "\<lbrakk>S \<subseteq> T; T \<subseteq> U\<rbrakk> \<Longrightarrow> (subtopology X U) interior_of S \<subseteq> (subtopology X T) interior_of S"
  by (metis dual_order.trans inf.orderE inf_commute interior_of_subset interior_of_subtopology_subsets)

lemma interior_of_subtopology_open:
  assumes "openin X U"
  shows "(subtopology X U) interior_of S = U \<inter> X interior_of S"
proof -
  have "\<forall>A. U \<inter> X closure_of (U \<inter> A) = U \<inter> X closure_of A"
    using assms openin_Int_closure_of_eq by blast
  then have "topspace X \<inter> U - U \<inter> X closure_of (topspace X \<inter> U - S) = U \<inter> (topspace X - X closure_of (topspace X - S))"
    by (metis (no_types) Diff_Int_distrib Int_Diff inf_commute)
  then show ?thesis
    unfolding interior_of_closure_of closure_of_subtopology_open topspace_subtopology
    using openin_Int_closure_of_eq [OF assms]
    by (metis assms closure_of_subtopology_open)
qed

lemma dense_intersects_open:
   "X closure_of S = topspace X \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<noteq> {} \<longrightarrow> S \<inter> T \<noteq> {})"
proof -
  have "X closure_of S = topspace X \<longleftrightarrow> (topspace X - X interior_of (topspace X - S) = topspace X)"
    by (simp add: closure_of_interior_of)
  also have "\<dots> \<longleftrightarrow> X interior_of (topspace X - S) = {}"
    by (simp add: closure_of_complement interior_of_eq_empty_complement)
  also have "\<dots> \<longleftrightarrow> (\<forall>T. openin X T \<and> T \<noteq> {} \<longrightarrow> S \<inter> T \<noteq> {})"
    unfolding interior_of_eq_empty_alt
    using openin_subset by fastforce
  finally show ?thesis .
qed

lemma interior_of_closedin_union_empty_interior_of:
  assumes "closedin X S" and disj: "X interior_of T = {}"
  shows "X interior_of (S \<union> T) = X interior_of S"
proof -
  have "X closure_of (topspace X - T) = topspace X"
    by (metis Diff_Diff_Int disj closure_of_eq_topspace closure_of_restrict interior_of_closure_of)
  then show ?thesis
    unfolding interior_of_closure_of
    by (metis Diff_Un Diff_subset assms(1) closedin_def closure_of_openin_Int_superset)
qed

lemma interior_of_union_eq_empty:
   "closedin X S
        \<Longrightarrow> (X interior_of (S \<union> T) = {} \<longleftrightarrow>
             X interior_of S = {} \<and> X interior_of T = {})"
  by (metis interior_of_closedin_union_empty_interior_of le_sup_iff subset_empty union_interior_of_subset)

lemma discrete_topology_interior_of [simp]:
    "(discrete_topology U) interior_of S = U \<inter> S"
  by (simp add: interior_of_restrict [of _ S] interior_of_eq)


subsection \<open>Frontier with respect to topological space \<close>

definition frontier_of :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixr "frontier'_of" 80)
  where "X frontier_of S \<equiv> X closure_of S - X interior_of S"

lemma frontier_of_closures:
     "X frontier_of S = X closure_of S \<inter> X closure_of (topspace X - S)"
  by (metis Diff_Diff_Int closure_of_complement closure_of_subset_topspace double_diff frontier_of_def interior_of_subset_closure_of)


lemma interior_of_union_frontier_of [simp]:
     "X interior_of S \<union> X frontier_of S = X closure_of S"
  by (simp add: frontier_of_def interior_of_subset_closure_of subset_antisym)

lemma frontier_of_restrict: "X frontier_of S = X frontier_of (topspace X \<inter> S)"
  by (metis closure_of_restrict frontier_of_def interior_of_restrict)

lemma closedin_frontier_of: "closedin X (X frontier_of S)"
  by (simp add: closedin_Int frontier_of_closures)

lemma frontier_of_subset_topspace: "X frontier_of S \<subseteq> topspace X"
  by (simp add: closedin_frontier_of closedin_subset)

lemma frontier_of_subset_subtopology: "(subtopology X S) frontier_of T \<subseteq> S"
  by (metis (no_types) closedin_derived_set closedin_frontier_of)

lemma frontier_of_subtopology_subset:
  "U \<inter> (subtopology X U) frontier_of S \<subseteq> (X frontier_of S)"
proof -
  have "U \<inter> X interior_of S - subtopology X U interior_of S = {}"
    by (simp add: interior_of_subtopology_subset)
  moreover have "X closure_of S \<inter> subtopology X U closure_of S = subtopology X U closure_of S"
    by (meson closure_of_subtopology_subset inf.absorb_iff2)
  ultimately show ?thesis
    unfolding frontier_of_def
    by blast
qed

lemma frontier_of_subtopology_mono:
   "\<lbrakk>S \<subseteq> T; T \<subseteq> U\<rbrakk> \<Longrightarrow> (subtopology X T) frontier_of S \<subseteq> (subtopology X U) frontier_of S"
    by (simp add: frontier_of_def Diff_mono closure_of_subtopology_mono interior_of_subtopology_mono)

lemma clopenin_eq_frontier_of:
   "closedin X S \<and> openin X S \<longleftrightarrow> S \<subseteq> topspace X \<and> X frontier_of S = {}"
proof (cases "S \<subseteq> topspace X")
  case True
  then show ?thesis
    by (metis Diff_eq_empty_iff closure_of_eq closure_of_subset_eq frontier_of_def interior_of_eq interior_of_subset interior_of_union_frontier_of sup_bot_right)
next
  case False
  then show ?thesis
    by (simp add: frontier_of_closures openin_closedin_eq)
qed

lemma frontier_of_eq_empty:
     "S \<subseteq> topspace X \<Longrightarrow> (X frontier_of S = {} \<longleftrightarrow> closedin X S \<and> openin X S)"
  by (simp add: clopenin_eq_frontier_of)

lemma frontier_of_openin:
     "openin X S \<Longrightarrow> X frontier_of S = X closure_of S - S"
  by (metis (no_types) frontier_of_def interior_of_eq)

lemma frontier_of_openin_straddle_Int:
  assumes "openin X U" "U \<inter> X frontier_of S \<noteq> {}"
  shows "U \<inter> S \<noteq> {}" "U - S \<noteq> {}"
proof -
  have "U \<inter> (X closure_of S \<inter> X closure_of (topspace X - S)) \<noteq> {}"
    using assms by (simp add: frontier_of_closures)
  then show "U \<inter> S \<noteq> {}"
    using assms openin_Int_closure_of_eq_empty by fastforce
  show "U - S \<noteq> {}"
  proof -
    have "\<exists>A. X closure_of (A - S) \<inter> U \<noteq> {}"
      using \<open>U \<inter> (X closure_of S \<inter> X closure_of (topspace X - S)) \<noteq> {}\<close> by blast
    then have "\<not> U \<subseteq> S"
      by (metis Diff_disjoint Diff_eq_empty_iff Int_Diff assms(1) inf_commute openin_Int_closure_of_eq_empty)
    then show ?thesis
      by blast
  qed
qed

lemma frontier_of_subset_closedin: "closedin X S \<Longrightarrow> (X frontier_of S) \<subseteq> S"
  using closure_of_eq frontier_of_def by fastforce

lemma frontier_of_empty [simp]: "X frontier_of {} = {}"
  by (simp add: frontier_of_def)

lemma frontier_of_topspace [simp]: "X frontier_of topspace X = {}"
  by (simp add: frontier_of_def)

lemma frontier_of_subset_eq:
  assumes "S \<subseteq> topspace X"
  shows "(X frontier_of S) \<subseteq> S \<longleftrightarrow> closedin X S"
proof
  show "X frontier_of S \<subseteq> S \<Longrightarrow> closedin X S"
    by (metis assms closure_of_subset_eq interior_of_subset interior_of_union_frontier_of le_sup_iff)
  show "closedin X S \<Longrightarrow> X frontier_of S \<subseteq> S"
    by (simp add: frontier_of_subset_closedin)
qed

lemma frontier_of_complement: "X frontier_of (topspace X - S) = X frontier_of S"
  by (metis Diff_Diff_Int closure_of_restrict frontier_of_closures inf_commute)

lemma frontier_of_disjoint_eq:
  assumes "S \<subseteq> topspace X"
  shows "((X frontier_of S) \<inter> S = {} \<longleftrightarrow> openin X S)"
proof
  assume "X frontier_of S \<inter> S = {}"
  then have "closedin X (topspace X - S)"
    using assms closure_of_subset frontier_of_def interior_of_eq interior_of_subset by fastforce
  then show "openin X S"
    using assms by (simp add: openin_closedin)
next
  show "openin X S \<Longrightarrow> X frontier_of S \<inter> S = {}"
    by (simp add: Diff_Diff_Int closedin_def frontier_of_openin inf.absorb_iff2 inf_commute)
qed

lemma frontier_of_disjoint_eq_alt:
  "S \<subseteq> (topspace X - X frontier_of S) \<longleftrightarrow> openin X S"
proof (cases "S \<subseteq> topspace X")
  case True
  show ?thesis
    using True frontier_of_disjoint_eq by auto
next
  case False
  then show ?thesis
    by (meson Diff_subset openin_subset subset_trans)
qed

lemma frontier_of_Int:
     "X frontier_of (S \<inter> T) =
      X closure_of (S \<inter> T) \<inter> (X frontier_of S \<union> X frontier_of T)"
proof -
  have *: "U \<subseteq> S \<and> U \<subseteq> T \<Longrightarrow> U \<inter> (S \<inter> A \<union> T \<inter> B) = U \<inter> (A \<union> B)" for U S T A B :: "'a set"
    by blast
  show ?thesis
    by (simp add: frontier_of_closures closure_of_mono Diff_Int * flip: closure_of_Un)
qed

lemma frontier_of_Int_subset: "X frontier_of (S \<inter> T) \<subseteq> X frontier_of S \<union> X frontier_of T"
  by (simp add: frontier_of_Int)

lemma frontier_of_Int_closedin:
  "\<lbrakk>closedin X S; closedin X T\<rbrakk> \<Longrightarrow> X frontier_of(S \<inter> T) = X frontier_of S \<inter> T \<union> S \<inter> X frontier_of T"
  apply (simp add: frontier_of_Int closedin_Int closure_of_closedin)
  using frontier_of_subset_closedin by blast

lemma frontier_of_Un_subset: "X frontier_of(S \<union> T) \<subseteq> X frontier_of S \<union> X frontier_of T"
  by (metis Diff_Un frontier_of_Int_subset frontier_of_complement)

lemma frontier_of_Union_subset:
   "finite \<F> \<Longrightarrow> X frontier_of (\<Union>\<F>) \<subseteq> (\<Union>T \<in> \<F>. X frontier_of T)"
proof (induction \<F> rule: finite_induct)
  case (insert A \<F>)
  then show ?case
    using frontier_of_Un_subset by fastforce
qed simp

lemma frontier_of_frontier_of_subset:
     "X frontier_of (X frontier_of S) \<subseteq> X frontier_of S"
  by (simp add: closedin_frontier_of frontier_of_subset_closedin)

lemma frontier_of_subtopology_open:
     "openin X U \<Longrightarrow> (subtopology X U) frontier_of S = U \<inter> X frontier_of S"
  by (simp add: Diff_Int_distrib closure_of_subtopology_open frontier_of_def interior_of_subtopology_open)

lemma discrete_topology_frontier_of [simp]:
     "(discrete_topology U) frontier_of S = {}"
  by (simp add: Diff_eq discrete_topology_closure_of frontier_of_closures)


subsection\<open>Continuous maps\<close>

definition continuous_map where
  "continuous_map X Y f \<equiv>
     (\<forall>x \<in> topspace X. f x \<in> topspace Y) \<and>
     (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"

lemma continuous_map:
   "continuous_map X Y f \<longleftrightarrow>
        f ` (topspace X) \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
  by (auto simp: continuous_map_def)

lemma continuous_map_image_subset_topspace:
   "continuous_map X Y f \<Longrightarrow> f ` (topspace X) \<subseteq> topspace Y"
  by (auto simp: continuous_map_def)

lemma continuous_map_on_empty: "topspace X = {} \<Longrightarrow> continuous_map X Y f"
  by (auto simp: continuous_map_def)

lemma continuous_map_closedin:
   "continuous_map X Y f \<longleftrightarrow>
         (\<forall>x \<in> topspace X. f x \<in> topspace Y) \<and>
         (\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C})"
proof -
  have "(\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U}) =
        (\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C})"
    if "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y"
  proof -
    have eq: "{x \<in> topspace X. f x \<in> topspace Y \<and> f x \<notin> C} = (topspace X - {x \<in> topspace X. f x \<in> C})" for C
      using that by blast
    show ?thesis
    proof (intro iffI allI impI)
      fix C
      assume "\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U}" and "closedin Y C"
      then have "openin X {x \<in> topspace X. f x \<in> topspace Y - C}" by blast
      then show "closedin X {x \<in> topspace X. f x \<in> C}"
        by (auto simp add: closedin_def eq)
    next
      fix U
      assume "\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C}" and "openin Y U"
      then have "closedin X {x \<in> topspace X. f x \<in> topspace Y - U}" by blast
      then show "openin X {x \<in> topspace X. f x \<in> U}"
        by (auto simp add: openin_closedin_eq eq)
    qed
  qed
  then show ?thesis
    by (auto simp: continuous_map_def)
qed

lemma openin_continuous_map_preimage:
   "\<lbrakk>continuous_map X Y f; openin Y U\<rbrakk> \<Longrightarrow> openin X {x \<in> topspace X. f x \<in> U}"
  by (simp add: continuous_map_def)

lemma closedin_continuous_map_preimage:
   "\<lbrakk>continuous_map X Y f; closedin Y C\<rbrakk> \<Longrightarrow> closedin X {x \<in> topspace X. f x \<in> C}"
  by (simp add: continuous_map_closedin)

lemma openin_continuous_map_preimage_gen:
  assumes "continuous_map X Y f" "openin X U" "openin Y V"
  shows "openin X {x \<in> U. f x \<in> V}"
proof -
  have eq: "{x \<in> U. f x \<in> V} = U \<inter> {x \<in> topspace X. f x \<in> V}"
    using assms(2) openin_closedin_eq by fastforce
  show ?thesis
    unfolding eq
    using assms openin_continuous_map_preimage by fastforce
qed

lemma closedin_continuous_map_preimage_gen:
  assumes "continuous_map X Y f" "closedin X U" "closedin Y V"
  shows "closedin X {x \<in> U. f x \<in> V}"
proof -
  have eq: "{x \<in> U. f x \<in> V} = U \<inter> {x \<in> topspace X. f x \<in> V}"
    using assms(2) closedin_def by fastforce
  show ?thesis
    unfolding eq
    using assms closedin_continuous_map_preimage by fastforce
qed

lemma continuous_map_image_closure_subset:
  assumes "continuous_map X Y f"
  shows "f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
proof -
  have *: "f ` (topspace X) \<subseteq> topspace Y"
    by (meson assms continuous_map)
  have "X closure_of T \<subseteq> {x \<in> X closure_of T. f x \<in> Y closure_of (f ` T)}" if "T \<subseteq> topspace X" for T
  proof (rule closure_of_minimal)
    show "T \<subseteq> {x \<in> X closure_of T. f x \<in> Y closure_of f ` T}"
      using closure_of_subset * that  by (fastforce simp: in_closure_of)
  next
    show "closedin X {x \<in> X closure_of T. f x \<in> Y closure_of f ` T}"
      using assms closedin_continuous_map_preimage_gen by fastforce
  qed
  then have "f ` (X closure_of (topspace X \<inter> S)) \<subseteq> Y closure_of (f ` (topspace X \<inter> S))"
    by blast
  also have "\<dots> \<subseteq> Y closure_of (topspace Y \<inter> f ` S)"
    using * by (blast intro!: closure_of_mono)
  finally have "f ` (X closure_of (topspace X \<inter> S)) \<subseteq> Y closure_of (topspace Y \<inter> f ` S)" .
  then show ?thesis
    by (metis closure_of_restrict)
qed

lemma continuous_map_subset_aux1: "continuous_map X Y f \<Longrightarrow>
       (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
  using continuous_map_image_closure_subset by blast

lemma continuous_map_subset_aux2:
  assumes "\<forall>S. S \<subseteq> topspace X \<longrightarrow> f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
  shows "continuous_map X Y f"
  unfolding continuous_map_closedin
proof (intro conjI ballI allI impI)
  fix x
  assume "x \<in> topspace X"
  then show "f x \<in> topspace Y"
    using assms closure_of_subset_topspace by fastforce
next
  fix C
  assume "closedin Y C"
  then show "closedin X {x \<in> topspace X. f x \<in> C}"
  proof (clarsimp simp flip: closure_of_subset_eq, intro conjI)
    fix x
    assume x: "x \<in> X closure_of {x \<in> topspace X. f x \<in> C}"
      and "C \<subseteq> topspace Y" and "Y closure_of C \<subseteq> C"
    show "x \<in> topspace X"
      by (meson x in_closure_of)
    have "{a \<in> topspace X. f a \<in> C} \<subseteq> topspace X"
      by simp
    moreover have "Y closure_of f ` {a \<in> topspace X. f a \<in> C} \<subseteq> C"
      by (simp add: \<open>closedin Y C\<close> closure_of_minimal image_subset_iff)
    ultimately have "f ` (X closure_of {a \<in> topspace X. f a \<in> C}) \<subseteq> C"
      using assms by blast
    then show "f x \<in> C"
      using x by auto
  qed
qed

lemma continuous_map_eq_image_closure_subset:
     "continuous_map X Y f \<longleftrightarrow> (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
  using continuous_map_subset_aux1 continuous_map_subset_aux2 by metis

lemma continuous_map_eq_image_closure_subset_alt:
     "continuous_map X Y f \<longleftrightarrow> (\<forall>S. S \<subseteq> topspace X \<longrightarrow> f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
  using continuous_map_subset_aux1 continuous_map_subset_aux2 by metis

lemma continuous_map_eq_image_closure_subset_gen:
     "continuous_map X Y f \<longleftrightarrow>
        f ` (topspace X) \<subseteq> topspace Y \<and>
        (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
  using continuous_map_subset_aux1 continuous_map_subset_aux2 continuous_map_image_subset_topspace by metis

lemma continuous_map_closure_preimage_subset:
   "continuous_map X Y f
        \<Longrightarrow> X closure_of {x \<in> topspace X. f x \<in> T}
            \<subseteq> {x \<in> topspace X. f x \<in> Y closure_of T}"
  unfolding continuous_map_closedin
  by (rule closure_of_minimal) (use in_closure_of in \<open>fastforce+\<close>)


lemma continuous_map_frontier_frontier_preimage_subset:
  assumes "continuous_map X Y f"
  shows "X frontier_of {x \<in> topspace X. f x \<in> T} \<subseteq> {x \<in> topspace X. f x \<in> Y frontier_of T}"
proof -
  have eq: "topspace X - {x \<in> topspace X. f x \<in> T} = {x \<in> topspace X. f x \<in> topspace Y - T}"
    using assms unfolding continuous_map_def by blast
  have "X closure_of {x \<in> topspace X. f x \<in> T} \<subseteq> {x \<in> topspace X. f x \<in> Y closure_of T}"
    by (simp add: assms continuous_map_closure_preimage_subset)
  moreover
  have "X closure_of (topspace X - {x \<in> topspace X. f x \<in> T}) \<subseteq> {x \<in> topspace X. f x \<in> Y closure_of (topspace Y - T)}"
    using continuous_map_closure_preimage_subset [OF assms] eq by presburger
  ultimately show ?thesis
    by (auto simp: frontier_of_closures)
qed

lemma continuous_map_id [simp]: "continuous_map X X id"
  unfolding continuous_map_def  using openin_subopen topspace_def by fastforce

lemma topology_finer_continuous_id:
  "topspace X = topspace Y \<Longrightarrow> ((\<forall>S. openin X S \<longrightarrow> openin Y S) \<longleftrightarrow> continuous_map Y X id)"
  unfolding continuous_map_def
  apply auto
  using openin_subopen openin_subset apply fastforce
  using openin_subopen topspace_def by fastforce

lemma continuous_map_const [simp]:
   "continuous_map X Y (\<lambda>x. C) \<longleftrightarrow> topspace X = {} \<or> C \<in> topspace Y"
proof (cases "topspace X = {}")
  case False
  show ?thesis
  proof (cases "C \<in> topspace Y")
    case True
    with openin_subopen show ?thesis
      by (auto simp: continuous_map_def)
  next
    case False
    then show ?thesis
      unfolding continuous_map_def by fastforce
  qed
qed (auto simp: continuous_map_on_empty)

lemma continuous_map_compose:
  assumes f: "continuous_map X X' f" and g: "continuous_map X' X'' g"
  shows "continuous_map X X'' (g \<circ> f)"
  unfolding continuous_map_def
proof (intro conjI ballI allI impI)
  fix x
  assume "x \<in> topspace X"
  then show "(g \<circ> f) x \<in> topspace X''"
    using assms unfolding continuous_map_def by force
next
  fix U
  assume "openin X'' U"
  have eq: "{x \<in> topspace X. (g \<circ> f) x \<in> U} = {x \<in> topspace X. f x \<in> {y. y \<in> topspace X' \<and> g y \<in> U}}"
    by auto (meson f continuous_map_def)
  show "openin X {x \<in> topspace X. (g \<circ> f) x \<in> U}"
    unfolding eq
    using assms unfolding continuous_map_def
    using \<open>openin X'' U\<close> by blast
qed

lemma continuous_map_eq:
  assumes "continuous_map X X' f" and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "continuous_map X X' g"
proof -
  have eq: "{x \<in> topspace X. f x \<in> U} = {x \<in> topspace X. g x \<in> U}" for U
    using assms by auto
  show ?thesis
    using assms by (simp add: continuous_map_def eq)
qed

lemma restrict_continuous_map [simp]:
     "topspace X \<subseteq> S \<Longrightarrow> continuous_map X X' (restrict f S) \<longleftrightarrow> continuous_map X X' f"
  by (auto simp: elim!: continuous_map_eq)

lemma continuous_map_in_subtopology:
  "continuous_map X (subtopology X' S) f \<longleftrightarrow> continuous_map X X' f \<and> f ` (topspace X) \<subseteq> S"
  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  show ?rhs
  proof -
    have "\<And>A. f ` (X closure_of A) \<subseteq> subtopology X' S closure_of f ` A"
      by (meson L continuous_map_image_closure_subset)
    then show ?thesis
      by (metis (no_types) closure_of_subset_subtopology closure_of_subtopology_subset closure_of_topspace continuous_map_eq_image_closure_subset dual_order.trans)
  qed
next
  assume R: ?rhs
  then have eq: "{x \<in> topspace X. f x \<in> U} = {x \<in> topspace X. f x \<in> U \<and> f x \<in> S}" for U
    by auto
  show ?lhs
    using R
    unfolding continuous_map
    by (auto simp: topspace_subtopology openin_subtopology eq)
qed


lemma continuous_map_from_subtopology:
     "continuous_map X X' f \<Longrightarrow> continuous_map (subtopology X S) X' f"
  by (auto simp: continuous_map topspace_subtopology openin_subtopology)

lemma continuous_map_into_fulltopology:
   "continuous_map X (subtopology X' T) f \<Longrightarrow> continuous_map X X' f"
  by (auto simp: continuous_map_in_subtopology)

lemma continuous_map_into_subtopology:
   "\<lbrakk>continuous_map X X' f; f ` topspace X \<subseteq> T\<rbrakk> \<Longrightarrow> continuous_map X (subtopology X' T) f"
  by (auto simp: continuous_map_in_subtopology)

lemma continuous_map_from_subtopology_mono:
     "\<lbrakk>continuous_map (subtopology X T) X' f; S \<subseteq> T\<rbrakk>
      \<Longrightarrow> continuous_map (subtopology X S) X' f"
  by (metis inf.absorb_iff2 continuous_map_from_subtopology subtopology_subtopology)

lemma continuous_map_from_discrete_topology [simp]:
  "continuous_map (discrete_topology U) X f \<longleftrightarrow> f ` U \<subseteq> topspace X"
  by (auto simp: continuous_map_def)

lemma continuous_map_iff_continuous_real [simp]: "continuous_map (subtopology euclideanreal S) euclideanreal g = continuous_on S g"
  by (force simp: continuous_map openin_subtopology continuous_on_open_invariant)


subsection\<open>Open and closed maps (not a priori assumed continuous)\<close>

definition open_map :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "open_map X1 X2 f \<equiv> \<forall>U. openin X1 U \<longrightarrow> openin X2 (f ` U)"

definition closed_map :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where "closed_map X1 X2 f \<equiv> \<forall>U. closedin X1 U \<longrightarrow> closedin X2 (f ` U)"

lemma open_map_imp_subset_topspace:
     "open_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"
  unfolding open_map_def by (simp add: openin_subset)

lemma open_map_imp_subset:
    "\<lbrakk>open_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f ` S \<subseteq> topspace X2"
  by (meson order_trans open_map_imp_subset_topspace subset_image_iff)

lemma topology_finer_open_id:
     "(\<forall>S. openin X S \<longrightarrow> openin X' S) \<longleftrightarrow> open_map X X' id"
  unfolding open_map_def by auto

lemma open_map_id: "open_map X X id"
  unfolding open_map_def by auto

lemma open_map_eq:
     "\<lbrakk>open_map X X' f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> open_map X X' g"
  unfolding open_map_def
  by (metis image_cong openin_subset subset_iff)

lemma open_map_inclusion_eq:
  "open_map (subtopology X S) X id \<longleftrightarrow> openin X (topspace X \<inter> S)"
proof -
  have *: "openin X (T \<inter> S)" if "openin X (S \<inter> topspace X)" "openin X T" for T
  proof -
    have "T \<subseteq> topspace X"
      using that by (simp add: openin_subset)
    with that show "openin X (T \<inter> S)"
      by (metis inf.absorb1 inf.left_commute inf_commute openin_Int)
  qed
  show ?thesis
    by (fastforce simp add: open_map_def Int_commute openin_subtopology_alt intro: *)
qed

lemma open_map_inclusion:
     "openin X S \<Longrightarrow> open_map (subtopology X S) X id"
  by (simp add: open_map_inclusion_eq openin_Int)

lemma open_map_compose:
     "\<lbrakk>open_map X X' f; open_map X' X'' g\<rbrakk> \<Longrightarrow> open_map X X'' (g \<circ> f)"
  by (metis (no_types, lifting) image_comp open_map_def)

lemma closed_map_imp_subset_topspace:
     "closed_map X1 X2 f \<Longrightarrow> f ` (topspace X1) \<subseteq> topspace X2"
  by (simp add: closed_map_def closedin_subset)

lemma closed_map_imp_subset:
     "\<lbrakk>closed_map X1 X2 f; S \<subseteq> topspace X1\<rbrakk> \<Longrightarrow> f ` S \<subseteq> topspace X2"
  using closed_map_imp_subset_topspace by blast

lemma topology_finer_closed_id:
    "(\<forall>S. closedin X S \<longrightarrow> closedin X' S) \<longleftrightarrow> closed_map X X' id"
  by (simp add: closed_map_def)

lemma closed_map_id: "closed_map X X id"
  by (simp add: closed_map_def)

lemma closed_map_eq:
   "\<lbrakk>closed_map X X' f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> closed_map X X' g"
  unfolding closed_map_def
  by (metis image_cong closedin_subset subset_iff)

lemma closed_map_compose:
    "\<lbrakk>closed_map X X' f; closed_map X' X'' g\<rbrakk> \<Longrightarrow> closed_map X X'' (g \<circ> f)"
  by (metis (no_types, lifting) closed_map_def image_comp)

lemma closed_map_inclusion_eq:
   "closed_map (subtopology X S) X id \<longleftrightarrow>
        closedin X (topspace X \<inter> S)"
proof -
  have *: "closedin X (T \<inter> S)" if "closedin X (S \<inter> topspace X)" "closedin X T" for T
  proof -
    have "T \<subseteq> topspace X"
      using that by (simp add: closedin_subset)
    with that show "closedin X (T \<inter> S)"
      by (metis inf.absorb1 inf.left_commute inf_commute closedin_Int)
  qed
  show ?thesis
    by (fastforce simp add: closed_map_def Int_commute closedin_subtopology_alt intro: *)
qed

lemma closed_map_inclusion: "closedin X S \<Longrightarrow> closed_map (subtopology X S) X id"
  by (simp add: closed_map_inclusion_eq closedin_Int)

lemma open_map_into_subtopology:
    "\<lbrakk>open_map X X' f; f ` topspace X \<subseteq> S\<rbrakk> \<Longrightarrow> open_map X (subtopology X' S) f"
  unfolding open_map_def openin_subtopology
  using openin_subset by fastforce

lemma closed_map_into_subtopology:
    "\<lbrakk>closed_map X X' f; f ` topspace X \<subseteq> S\<rbrakk> \<Longrightarrow> closed_map X (subtopology X' S) f"
  unfolding closed_map_def closedin_subtopology
  using closedin_subset by fastforce

lemma open_map_into_discrete_topology:
    "open_map X (discrete_topology U) f \<longleftrightarrow> f ` (topspace X) \<subseteq> U"
  unfolding open_map_def openin_discrete_topology using openin_subset by blast

lemma closed_map_into_discrete_topology:
    "closed_map X (discrete_topology U) f \<longleftrightarrow> f ` (topspace X) \<subseteq> U"
  unfolding closed_map_def closedin_discrete_topology using closedin_subset by blast

lemma bijective_open_imp_closed_map:
     "\<lbrakk>open_map X X' f; f ` (topspace X) = topspace X'; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> closed_map X X' f"
  unfolding open_map_def closed_map_def closedin_def
  by auto (metis Diff_subset inj_on_image_set_diff)

lemma bijective_closed_imp_open_map:
     "\<lbrakk>closed_map X X' f; f ` (topspace X) = topspace X'; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> open_map X X' f"
  unfolding closed_map_def open_map_def openin_closedin_eq
  by auto (metis Diff_subset inj_on_image_set_diff)

lemma open_map_from_subtopology:
     "\<lbrakk>open_map X X' f; openin X U\<rbrakk> \<Longrightarrow> open_map (subtopology X U) X' f"
  unfolding open_map_def openin_subtopology_alt by blast

lemma closed_map_from_subtopology:
     "\<lbrakk>closed_map X X' f; closedin X U\<rbrakk> \<Longrightarrow> closed_map (subtopology X U) X' f"
  unfolding closed_map_def closedin_subtopology_alt by blast

lemma open_map_restriction:
     "\<lbrakk>open_map X X' f; {x. x \<in> topspace X \<and> f x \<in> V} = U\<rbrakk>
      \<Longrightarrow> open_map (subtopology X U) (subtopology X' V) f"
  unfolding open_map_def openin_subtopology_alt
  apply clarify
  apply (rename_tac T)
  apply (rule_tac x="f ` T" in image_eqI)
  using openin_closedin_eq by force+

lemma closed_map_restriction:
     "\<lbrakk>closed_map X X' f; {x. x \<in> topspace X \<and> f x \<in> V} = U\<rbrakk>
      \<Longrightarrow> closed_map (subtopology X U) (subtopology X' V) f"
  unfolding closed_map_def closedin_subtopology_alt
  apply clarify
  apply (rename_tac T)
  apply (rule_tac x="f ` T" in image_eqI)
  using closedin_def by force+

subsection\<open>Quotient maps\<close>
                                      
definition quotient_map where
 "quotient_map X X' f \<longleftrightarrow>
        f ` (topspace X) = topspace X' \<and>
        (\<forall>U. U \<subseteq> topspace X' \<longrightarrow> (openin X {x. x \<in> topspace X \<and> f x \<in> U} \<longleftrightarrow> openin X' U))"

lemma quotient_map_eq:
  assumes "quotient_map X X' f" "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x"
  shows "quotient_map X X' g"
proof -
  have eq: "{x \<in> topspace X. f x \<in> U} = {x \<in> topspace X. g x \<in> U}" for U
    using assms by auto
  show ?thesis
  using assms
  unfolding quotient_map_def
  by (metis (mono_tags, lifting) eq image_cong)
qed

lemma quotient_map_compose:
  assumes f: "quotient_map X X' f" and g: "quotient_map X' X'' g"
  shows "quotient_map X X'' (g \<circ> f)"
  unfolding quotient_map_def
proof (intro conjI allI impI)
  show "(g \<circ> f) ` topspace X = topspace X''"
    using assms by (simp only: image_comp [symmetric]) (simp add: quotient_map_def)
next
  fix U''
  assume "U'' \<subseteq> topspace X''"
  define U' where "U' \<equiv> {y \<in> topspace X'. g y \<in> U''}"
  have "U' \<subseteq> topspace X'"
    by (auto simp add: U'_def)
  then have U': "openin X {x \<in> topspace X. f x \<in> U'} = openin X' U'"
    using assms unfolding quotient_map_def by simp
  have eq: "{x \<in> topspace X. f x \<in> topspace X' \<and> g (f x) \<in> U''} = {x \<in> topspace X. (g \<circ> f) x \<in> U''}"
    using f quotient_map_def by fastforce
  have "openin X {x \<in> topspace X. (g \<circ> f) x \<in> U''} = openin X {x \<in> topspace X. f x \<in> U'}"
    using assms  by (simp add: quotient_map_def U'_def eq)
  also have "\<dots> = openin X'' U''"
    using U'_def \<open>U'' \<subseteq> topspace X''\<close> U' g quotient_map_def by fastforce
  finally show "openin X {x \<in> topspace X. (g \<circ> f) x \<in> U''} = openin X'' U''" .
qed

lemma quotient_map_from_composition:
  assumes f: "continuous_map X X' f" and g: "continuous_map X' X'' g" and gf: "quotient_map X X'' (g \<circ> f)"
  shows  "quotient_map X' X'' g"
  unfolding quotient_map_def
proof (intro conjI allI impI)
  show "g ` topspace X' = topspace X''"
    using assms unfolding continuous_map_def quotient_map_def by fastforce
next
  fix U'' :: "'c set"
  assume U'': "U'' \<subseteq> topspace X''"
  have eq: "{x \<in> topspace X. g (f x) \<in> U''} = {x \<in> topspace X. f x \<in> {y. y \<in> topspace X' \<and> g y \<in> U''}}"
    using continuous_map_def f by fastforce
  show "openin X' {x \<in> topspace X'. g x \<in> U''} = openin X'' U''"
    using assms unfolding continuous_map_def quotient_map_def
    by (metis (mono_tags, lifting) Collect_cong U'' comp_apply eq)
qed

lemma quotient_imp_continuous_map:
    "quotient_map X X' f \<Longrightarrow> continuous_map X X' f"
  by (simp add: continuous_map openin_subset quotient_map_def)

lemma quotient_imp_surjective_map:
    "quotient_map X X' f \<Longrightarrow> f ` (topspace X) = topspace X'"
  by (simp add: quotient_map_def)

lemma quotient_map_closedin:
  "quotient_map X X' f \<longleftrightarrow>
        f ` (topspace X) = topspace X' \<and>
        (\<forall>U. U \<subseteq> topspace X' \<longrightarrow> (closedin X {x. x \<in> topspace X \<and> f x \<in> U} \<longleftrightarrow> closedin X' U))"
proof -
  have eq: "(topspace X - {x \<in> topspace X. f x \<in> U'}) = {x \<in> topspace X. f x \<in> topspace X' \<and> f x \<notin> U'}"
    if "f ` topspace X = topspace X'" "U' \<subseteq> topspace X'" for U'
      using that by auto
  have "(\<forall>U\<subseteq>topspace X'. openin X {x \<in> topspace X. f x \<in> U} = openin X' U) =
          (\<forall>U\<subseteq>topspace X'. closedin X {x \<in> topspace X. f x \<in> U} = closedin X' U)"
    if "f ` topspace X = topspace X'"
  proof (rule iffI; intro allI impI subsetI)
    fix U'
    assume *[rule_format]: "\<forall>U\<subseteq>topspace X'. openin X {x \<in> topspace X. f x \<in> U} = openin X' U"
      and U': "U' \<subseteq> topspace X'"
    show "closedin X {x \<in> topspace X. f x \<in> U'} = closedin X' U'"
      using U'  by (auto simp add: closedin_def simp flip: * [of "topspace X' - U'"] eq [OF that])
  next
    fix U' :: "'b set"
    assume *[rule_format]: "\<forall>U\<subseteq>topspace X'. closedin X {x \<in> topspace X. f x \<in> U} = closedin X' U"
      and U': "U' \<subseteq> topspace X'"
    show "openin X {x \<in> topspace X. f x \<in> U'} = openin X' U'"
      using U'  by (auto simp add: openin_closedin_eq simp flip: * [of "topspace X' - U'"] eq [OF that])
  qed
  then show ?thesis
    unfolding quotient_map_def by force
qed

lemma continuous_open_imp_quotient_map:
  assumes "continuous_map X X' f" and om: "open_map X X' f" and feq: "f ` (topspace X) = topspace X'"
  shows "quotient_map X X' f"
proof -
  { fix U
    assume U: "U \<subseteq> topspace X'" and "openin X {x \<in> topspace X. f x \<in> U}"
    then have ope: "openin X' (f ` {x \<in> topspace X. f x \<in> U})"
      using om unfolding open_map_def by blast
    then have "openin X' U"
      using U feq by (subst openin_subopen) force
  }
  moreover have "openin X {x \<in> topspace X. f x \<in> U}" if "U \<subseteq> topspace X'" and "openin X' U" for U
    using that assms unfolding continuous_map_def by blast
  ultimately show ?thesis
    unfolding quotient_map_def using assms by blast
qed

lemma continuous_closed_imp_quotient_map:
  assumes "continuous_map X X' f" and om: "closed_map X X' f" and feq: "f ` (topspace X) = topspace X'"
  shows "quotient_map X X' f"
proof -
  have "f ` {x \<in> topspace X. f x \<in> U} = U" if "U \<subseteq> topspace X'" for U
    using that feq by auto
  with assms show ?thesis
    unfolding quotient_map_closedin closed_map_def continuous_map_closedin by auto
qed

lemma continuous_open_quotient_map:
   "\<lbrakk>continuous_map X X' f; open_map X X' f\<rbrakk> \<Longrightarrow> quotient_map X X' f \<longleftrightarrow> f ` (topspace X) = topspace X'"
  by (meson continuous_open_imp_quotient_map quotient_map_def)

lemma continuous_closed_quotient_map:
     "\<lbrakk>continuous_map X X' f; closed_map X X' f\<rbrakk> \<Longrightarrow> quotient_map X X' f \<longleftrightarrow> f ` (topspace X) = topspace X'"
  by (meson continuous_closed_imp_quotient_map quotient_map_def)

lemma injective_quotient_map:
  assumes "inj_on f (topspace X)"
  shows "quotient_map X X' f \<longleftrightarrow>
         continuous_map X X' f \<and> open_map X X' f \<and> closed_map X X' f \<and> f ` (topspace X) = topspace X'"
         (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  have "open_map X X' f"
  proof (clarsimp simp add: open_map_def)
    fix U
    assume "openin X U"
    then have "U \<subseteq> topspace X"
      by (simp add: openin_subset)
    moreover have "{x \<in> topspace X. f x \<in> f ` U} = U"
      using \<open>U \<subseteq> topspace X\<close> assms inj_onD by fastforce
    ultimately show "openin X' (f ` U)"
      using L unfolding quotient_map_def
      by (metis (no_types, lifting) Collect_cong \<open>openin X U\<close> image_mono)
  qed
  moreover have "closed_map X X' f"
  proof (clarsimp simp add: closed_map_def)
    fix U
    assume "closedin X U"
    then have "U \<subseteq> topspace X"
      by (simp add: closedin_subset)
    moreover have "{x \<in> topspace X. f x \<in> f ` U} = U"
      using \<open>U \<subseteq> topspace X\<close> assms inj_onD by fastforce
    ultimately show "closedin X' (f ` U)"
      using L unfolding quotient_map_closedin
      by (metis (no_types, lifting) Collect_cong \<open>closedin X U\<close> image_mono)
  qed
  ultimately show ?rhs
    using L by (simp add: quotient_imp_continuous_map quotient_imp_surjective_map)
next
  assume ?rhs
  then show ?lhs
    by (simp add: continuous_closed_imp_quotient_map)
qed

lemma continuous_compose_quotient_map:
  assumes f: "quotient_map X X' f" and g: "continuous_map X X'' (g \<circ> f)"
  shows "continuous_map X' X'' g"
  unfolding quotient_map_def continuous_map_def
proof (intro conjI ballI allI impI)
  show "\<And>x'. x' \<in> topspace X' \<Longrightarrow> g x' \<in> topspace X''"
    using assms unfolding quotient_map_def
    by (metis (no_types, hide_lams) continuous_map_image_subset_topspace image_comp image_subset_iff)
next
  fix U'' :: "'c set"
  assume U'': "openin X'' U''"
  have "f ` topspace X = topspace X'"
    by (simp add: f quotient_imp_surjective_map)
  then have eq: "{x \<in> topspace X. f x \<in> topspace X' \<and> g (f x) \<in> U} = {x \<in> topspace X. g (f x) \<in> U}" for U
    by auto
  have "openin X {x \<in> topspace X. f x \<in> topspace X' \<and> g (f x) \<in> U''}"
    unfolding eq using U'' g openin_continuous_map_preimage by fastforce
  then have *: "openin X {x \<in> topspace X. f x \<in> {x \<in> topspace X'. g x \<in> U''}}"
    by auto
  show "openin X' {x \<in> topspace X'. g x \<in> U''}"
    using f unfolding quotient_map_def
    by (metis (no_types) Collect_subset *)
qed

lemma continuous_compose_quotient_map_eq:
   "quotient_map X X' f \<Longrightarrow> continuous_map X X'' (g \<circ> f) \<longleftrightarrow> continuous_map X' X'' g"
  using continuous_compose_quotient_map continuous_map_compose quotient_imp_continuous_map by blast

lemma quotient_map_compose_eq:
   "quotient_map X X' f \<Longrightarrow> quotient_map X X'' (g \<circ> f) \<longleftrightarrow> quotient_map X' X'' g"
  apply safe
  apply (meson continuous_compose_quotient_map_eq quotient_imp_continuous_map quotient_map_from_composition)
  by (simp add: quotient_map_compose)

lemma quotient_map_restriction:
  assumes quo: "quotient_map X Y f" and U: "{x \<in> topspace X. f x \<in> V} = U" and disj: "openin Y V \<or> closedin Y V"
 shows "quotient_map (subtopology X U) (subtopology Y V) f"
  using disj
proof
  assume V: "openin Y V"
  with U have sub: "U \<subseteq> topspace X" "V \<subseteq> topspace Y"
    by (auto simp: openin_subset)
  have fim: "f ` topspace X = topspace Y"
     and Y: "\<And>U. U \<subseteq> topspace Y \<Longrightarrow> openin X {x \<in> topspace X. f x \<in> U} = openin Y U"
    using quo unfolding quotient_map_def by auto
  have "openin X U"
    using U V Y sub(2) by blast
  show ?thesis
    unfolding quotient_map_def
  proof (intro conjI allI impI)
    show "f ` topspace (subtopology X U) = topspace (subtopology Y V)"
      using sub U fim by (auto simp: topspace_subtopology)
  next
    fix Y' :: "'b set"
    assume "Y' \<subseteq> topspace (subtopology Y V)"
    then have "Y' \<subseteq> topspace Y" "Y' \<subseteq> V"
      by (simp_all add: topspace_subtopology)
    then have eq: "{x \<in> topspace X. x \<in> U \<and> f x \<in> Y'} = {x \<in> topspace X. f x \<in> Y'}"
      using U by blast
    then show "openin (subtopology X U) {x \<in> topspace (subtopology X U). f x \<in> Y'} = openin (subtopology Y V) Y'"
      using U V Y \<open>openin X U\<close>  \<open>Y' \<subseteq> topspace Y\<close> \<open>Y' \<subseteq> V\<close>
      by (simp add: topspace_subtopology openin_open_subtopology eq) (auto simp: openin_closedin_eq)
  qed
next
  assume V: "closedin Y V"
  with U have sub: "U \<subseteq> topspace X" "V \<subseteq> topspace Y"
    by (auto simp: closedin_subset)
  have fim: "f ` topspace X = topspace Y"
     and Y: "\<And>U. U \<subseteq> topspace Y \<Longrightarrow> closedin X {x \<in> topspace X. f x \<in> U} = closedin Y U"
    using quo unfolding quotient_map_closedin by auto
  have "closedin X U"
    using U V Y sub(2) by blast
  show ?thesis
    unfolding quotient_map_closedin
  proof (intro conjI allI impI)
    show "f ` topspace (subtopology X U) = topspace (subtopology Y V)"
      using sub U fim by (auto simp: topspace_subtopology)
  next
    fix Y' :: "'b set"
    assume "Y' \<subseteq> topspace (subtopology Y V)"
    then have "Y' \<subseteq> topspace Y" "Y' \<subseteq> V"
      by (simp_all add: topspace_subtopology)
    then have eq: "{x \<in> topspace X. x \<in> U \<and> f x \<in> Y'} = {x \<in> topspace X. f x \<in> Y'}"
      using U by blast
    then show "closedin (subtopology X U) {x \<in> topspace (subtopology X U). f x \<in> Y'} = closedin (subtopology Y V) Y'"
      using U V Y \<open>closedin X U\<close>  \<open>Y' \<subseteq> topspace Y\<close> \<open>Y' \<subseteq> V\<close>
      by (simp add: topspace_subtopology closedin_closed_subtopology eq) (auto simp: closedin_def)
  qed
qed

lemma quotient_map_saturated_open:
     "quotient_map X Y f \<longleftrightarrow>
        continuous_map X Y f \<and> f ` (topspace X) = topspace Y \<and>
        (\<forall>U. openin X U \<and> {x \<in> topspace X. f x \<in> f ` U} \<subseteq> U \<longrightarrow> openin Y (f ` U))"
     (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then have fim: "f ` topspace X = topspace Y"
    and Y: "\<And>U. U \<subseteq> topspace Y \<Longrightarrow> openin Y U = openin X {x \<in> topspace X. f x \<in> U}"
    unfolding quotient_map_def by auto
  show ?rhs
  proof (intro conjI allI impI)
    show "continuous_map X Y f"
      by (simp add: L quotient_imp_continuous_map)
    show "f ` topspace X = topspace Y"
      by (simp add: fim)
  next
    fix U :: "'a set"
    assume U: "openin X U \<and> {x \<in> topspace X. f x \<in> f ` U} \<subseteq> U"
    then have sub:  "f ` U \<subseteq> topspace Y" and eq: "{x \<in> topspace X. f x \<in> f ` U} = U"
      using fim openin_subset by fastforce+
    show "openin Y (f ` U)"
      by (simp add: sub Y eq U)
  qed
next
  assume ?rhs
  then have YX: "\<And>U. openin Y U \<Longrightarrow> openin X {x \<in> topspace X. f x \<in> U}"
       and fim: "f ` topspace X = topspace Y"
       and XY: "\<And>U. \<lbrakk>openin X U; {x \<in> topspace X. f x \<in> f ` U} \<subseteq> U\<rbrakk> \<Longrightarrow> openin Y (f ` U)"
    by (auto simp: quotient_map_def continuous_map_def)
  show ?lhs
  proof (simp add: quotient_map_def fim, intro allI impI iffI)
    fix U :: "'b set"
    assume "U \<subseteq> topspace Y" and X: "openin X {x \<in> topspace X. f x \<in> U}"
    have feq: "f ` {x \<in> topspace X. f x \<in> U} = U"
      using \<open>U \<subseteq> topspace Y\<close> fim by auto
    show "openin Y U"
      using XY [OF X] by (simp add: feq)
  next
    fix U :: "'b set"
    assume "U \<subseteq> topspace Y" and Y: "openin Y U"
    show "openin X {x \<in> topspace X. f x \<in> U}"
      by (metis YX [OF Y])
  qed
qed

subsection\<open> Separated Sets\<close>

definition separatedin :: "'a topology \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"
  where "separatedin X S T \<equiv>
           S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and>
           S \<inter> X closure_of T = {} \<and> T \<inter> X closure_of S = {}"

lemma separatedin_empty [simp]:
     "separatedin X S {} \<longleftrightarrow> S \<subseteq> topspace X"
     "separatedin X {} S \<longleftrightarrow> S \<subseteq> topspace X"
  by (simp_all add: separatedin_def)

lemma separatedin_refl [simp]:
     "separatedin X S S \<longleftrightarrow> S = {}"
proof -
  have "\<And>x. \<lbrakk>separatedin X S S; x \<in> S\<rbrakk> \<Longrightarrow> False"
    by (metis all_not_in_conv closure_of_subset inf.orderE separatedin_def)
  then show ?thesis
    by auto
qed

lemma separatedin_sym:
     "separatedin X S T \<longleftrightarrow> separatedin X T S"
  by (auto simp: separatedin_def)

lemma separatedin_imp_disjoint:
     "separatedin X S T \<Longrightarrow> disjnt S T"
  by (meson closure_of_subset disjnt_def disjnt_subset2 separatedin_def)

lemma separatedin_mono:
   "\<lbrakk>separatedin X S T; S' \<subseteq> S; T' \<subseteq> T\<rbrakk> \<Longrightarrow> separatedin X S' T'"
  unfolding separatedin_def
  using closure_of_mono by blast

lemma separatedin_open_sets:
     "\<lbrakk>openin X S; openin X T\<rbrakk> \<Longrightarrow> separatedin X S T \<longleftrightarrow> disjnt S T"
  unfolding disjnt_def separatedin_def
  by (auto simp: openin_Int_closure_of_eq_empty openin_subset)

lemma separatedin_closed_sets:
     "\<lbrakk>closedin X S; closedin X T\<rbrakk> \<Longrightarrow> separatedin X S T \<longleftrightarrow> disjnt S T"
  by (metis closedin_def closure_of_eq disjnt_def inf_commute separatedin_def)

lemma separatedin_subtopology:
     "separatedin (subtopology X U) S T \<longleftrightarrow> S \<subseteq> U \<and> T \<subseteq> U \<and> separatedin X S T"
  apply (simp add: separatedin_def closure_of_subtopology topspace_subtopology)
  apply (safe; metis Int_absorb1 inf.assoc inf.orderE insert_disjoint(2) mk_disjoint_insert)
  done

lemma separatedin_discrete_topology:
     "separatedin (discrete_topology U) S T \<longleftrightarrow> S \<subseteq> U \<and> T \<subseteq> U \<and> disjnt S T"
  by (metis openin_discrete_topology separatedin_def separatedin_open_sets topspace_discrete_topology)

lemma separated_eq_distinguishable:
   "separatedin X {x} {y} \<longleftrightarrow>
        x \<in> topspace X \<and> y \<in> topspace X \<and>
        (\<exists>U. openin X U \<and> x \<in> U \<and> (y \<notin> U)) \<and>
        (\<exists>v. openin X v \<and> y \<in> v \<and> (x \<notin> v))"
  by (force simp: separatedin_def closure_of_def)

lemma separatedin_Un [simp]:
   "separatedin X S (T \<union> U) \<longleftrightarrow> separatedin X S T \<and> separatedin X S U"
   "separatedin X (S \<union> T) U \<longleftrightarrow> separatedin X S U \<and> separatedin X T U"
  by (auto simp: separatedin_def)

lemma separatedin_Union:
  "finite \<F> \<Longrightarrow> separatedin X S (\<Union>\<F>) \<longleftrightarrow> S \<subseteq> topspace X \<and> (\<forall>T \<in> \<F>. separatedin X S T)"
  "finite \<F> \<Longrightarrow> separatedin X (\<Union>\<F>) S \<longleftrightarrow> (\<forall>T \<in> \<F>. separatedin X S T) \<and> S \<subseteq> topspace X"
  by (auto simp: separatedin_def closure_of_Union)

lemma separatedin_openin_diff:
   "\<lbrakk>openin X S; openin X T\<rbrakk> \<Longrightarrow> separatedin X (S - T) (T - S)"
  unfolding separatedin_def
  apply (intro conjI)
  apply (meson Diff_subset openin_subset subset_trans)+
  using openin_Int_closure_of_eq_empty by fastforce+

lemma separatedin_closedin_diff:
     "\<lbrakk>closedin X S; closedin X T\<rbrakk> \<Longrightarrow> separatedin X (S - T) (T - S)"
  apply (simp add: separatedin_def Diff_Int_distrib2 closure_of_minimal inf_absorb2)
  apply (meson Diff_subset closedin_subset subset_trans)
  done

lemma separation_closedin_Un_gen:
     "separatedin X S T \<longleftrightarrow>
        S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> disjnt S T \<and>
        closedin (subtopology X (S \<union> T)) S \<and>
        closedin (subtopology X (S \<union> T)) T"
  apply (simp add: separatedin_def closedin_Int_closure_of disjnt_iff)
  using closure_of_subset apply blast
  done

lemma separation_openin_Un_gen:
     "separatedin X S T \<longleftrightarrow>
        S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> disjnt S T \<and>
        openin (subtopology X (S \<union> T)) S \<and>
        openin (subtopology X (S \<union> T)) T"
  unfolding openin_closedin_eq topspace_subtopology separation_closedin_Un_gen disjnt_def
  by (auto simp: Diff_triv Int_commute Un_Diff inf_absorb1 topspace_def)


subsection\<open>Homeomorphisms\<close>
text\<open>(1-way and 2-way versions may be useful in places)\<close>

definition homeomorphic_map :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
  where
 "homeomorphic_map X Y f \<equiv> quotient_map X Y f \<and> inj_on f (topspace X)"

definition homeomorphic_maps :: "'a topology \<Rightarrow> 'b topology \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"
  where
 "homeomorphic_maps X Y f g \<equiv>
    continuous_map X Y f \<and> continuous_map Y X g \<and>
     (\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)"


lemma homeomorphic_map_eq:
   "\<lbrakk>homeomorphic_map X Y f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> homeomorphic_map X Y g"
  by (meson homeomorphic_map_def inj_on_cong quotient_map_eq)

lemma homeomorphic_maps_eq:
     "\<lbrakk>homeomorphic_maps X Y f g;
       \<And>x. x \<in> topspace X \<Longrightarrow> f x = f' x; \<And>y. y \<in> topspace Y \<Longrightarrow> g y = g' y\<rbrakk>
      \<Longrightarrow> homeomorphic_maps X Y f' g'"
  apply (simp add: homeomorphic_maps_def)
  by (metis continuous_map_eq continuous_map_eq_image_closure_subset_gen image_subset_iff)

lemma homeomorphic_maps_sym:
     "homeomorphic_maps X Y f g \<longleftrightarrow> homeomorphic_maps Y X g f"
  by (auto simp: homeomorphic_maps_def)

lemma homeomorphic_maps_id:
     "homeomorphic_maps X Y id id \<longleftrightarrow> Y = X"
  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then have "topspace X = topspace Y"
    by (auto simp: homeomorphic_maps_def continuous_map_def)
  with L show ?rhs
    unfolding homeomorphic_maps_def
    by (metis topology_finer_continuous_id topology_eq)
next
  assume ?rhs
  then show ?lhs
    unfolding homeomorphic_maps_def by auto
qed

lemma homeomorphic_map_id [simp]: "homeomorphic_map X Y id \<longleftrightarrow> Y = X"
       (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then have eq: "topspace X = topspace Y"
    by (auto simp: homeomorphic_map_def continuous_map_def quotient_map_def)
  then have "\<And>S. openin X S \<longrightarrow> openin Y S"
    by (meson L homeomorphic_map_def injective_quotient_map topology_finer_open_id)
  then show ?rhs
    using L unfolding homeomorphic_map_def
    by (metis eq quotient_imp_continuous_map topology_eq topology_finer_continuous_id)
next
  assume ?rhs
  then show ?lhs
    unfolding homeomorphic_map_def
    by (simp add: closed_map_id continuous_closed_imp_quotient_map)
qed

lemma homeomorphic_maps_i [simp]:"homeomorphic_maps X Y id id \<longleftrightarrow> Y = X"
  by (metis (full_types) eq_id_iff homeomorphic_maps_id)

lemma homeomorphic_map_i [simp]: "homeomorphic_map X Y id \<longleftrightarrow> Y = X"
  by (metis (no_types) eq_id_iff homeomorphic_map_id)

lemma homeomorphic_map_compose:
  assumes "homeomorphic_map X Y f" "homeomorphic_map Y X'' g"
  shows "homeomorphic_map X X'' (g \<circ> f)"
proof -
  have "inj_on g (f ` topspace X)"
    by (metis (no_types) assms homeomorphic_map_def quotient_imp_surjective_map)
  then show ?thesis
    using assms by (meson comp_inj_on homeomorphic_map_def quotient_map_compose_eq)
qed

lemma homeomorphic_maps_compose:
   "homeomorphic_maps X Y f h \<and>
        homeomorphic_maps Y X'' g k
        \<Longrightarrow> homeomorphic_maps X X'' (g \<circ> f) (h \<circ> k)"
  unfolding homeomorphic_maps_def
  by (auto simp: continuous_map_compose; simp add: continuous_map_def)

lemma homeomorphic_eq_everything_map:
   "homeomorphic_map X Y f \<longleftrightarrow>
        continuous_map X Y f \<and> open_map X Y f \<and> closed_map X Y f \<and>
        f ` (topspace X) = topspace Y \<and> inj_on f (topspace X)"
  unfolding homeomorphic_map_def
  by (force simp: injective_quotient_map intro: injective_quotient_map)

lemma homeomorphic_imp_continuous_map:
     "homeomorphic_map X Y f \<Longrightarrow> continuous_map X Y f"
  by (simp add: homeomorphic_eq_everything_map)

lemma homeomorphic_imp_open_map:
   "homeomorphic_map X Y f \<Longrightarrow> open_map X Y f"
  by (simp add: homeomorphic_eq_everything_map)

lemma homeomorphic_imp_closed_map:
   "homeomorphic_map X Y f \<Longrightarrow> closed_map X Y f"
  by (simp add: homeomorphic_eq_everything_map)

lemma homeomorphic_imp_surjective_map:
   "homeomorphic_map X Y f \<Longrightarrow> f ` (topspace X) = topspace Y"
  by (simp add: homeomorphic_eq_everything_map)

lemma homeomorphic_imp_injective_map:
    "homeomorphic_map X Y f \<Longrightarrow> inj_on f (topspace X)"
  by (simp add: homeomorphic_eq_everything_map)

lemma bijective_open_imp_homeomorphic_map:
   "\<lbrakk>continuous_map X Y f; open_map X Y f; f ` (topspace X) = topspace Y; inj_on f (topspace X)\<rbrakk>
        \<Longrightarrow> homeomorphic_map X Y f"
  by (simp add: homeomorphic_map_def continuous_open_imp_quotient_map)

lemma bijective_closed_imp_homeomorphic_map:
   "\<lbrakk>continuous_map X Y f; closed_map X Y f; f ` (topspace X) = topspace Y; inj_on f (topspace X)\<rbrakk>
        \<Longrightarrow> homeomorphic_map X Y f"
  by (simp add: continuous_closed_quotient_map homeomorphic_map_def)

lemma open_eq_continuous_inverse_map:
  assumes X: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y \<and> g(f x) = x"
    and Y: "\<And>y. y \<in> topspace Y \<Longrightarrow> g y \<in> topspace X \<and> f(g y) = y"
  shows "open_map X Y f \<longleftrightarrow> continuous_map Y X g"
proof -
  have eq: "{x \<in> topspace Y. g x \<in> U} = f ` U" if "openin X U" for U
    using openin_subset [OF that] by (force simp: X Y image_iff)
  show ?thesis
    by (auto simp: Y open_map_def continuous_map_def eq)
qed

lemma closed_eq_continuous_inverse_map:
  assumes X: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y \<and> g(f x) = x"
    and Y: "\<And>y. y \<in> topspace Y \<Longrightarrow> g y \<in> topspace X \<and> f(g y) = y"
  shows "closed_map X Y f \<longleftrightarrow> continuous_map Y X g"
proof -
  have eq: "{x \<in> topspace Y. g x \<in> U} = f ` U" if "closedin X U" for U
    using closedin_subset [OF that] by (force simp: X Y image_iff)
  show ?thesis
    by (auto simp: Y closed_map_def continuous_map_closedin eq)
qed

lemma homeomorphic_maps_map:
  "homeomorphic_maps X Y f g \<longleftrightarrow>
        homeomorphic_map X Y f \<and> homeomorphic_map Y X g \<and>
        (\<forall>x \<in> topspace X. g(f x) = x) \<and> (\<forall>y \<in> topspace Y. f(g y) = y)"
  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have L: "continuous_map X Y f" "continuous_map Y X g" "\<forall>x\<in>topspace X. g (f x) = x" "\<forall>x'\<in>topspace Y. f (g x') = x'"
    by (auto simp: homeomorphic_maps_def)
  show ?rhs
  proof (intro conjI bijective_open_imp_homeomorphic_map L)
    show "open_map X Y f"
      using L using open_eq_continuous_inverse_map [of concl: X Y f g] by (simp add: continuous_map_def)
    show "open_map Y X g"
      using L using open_eq_continuous_inverse_map [of concl: Y X g f] by (simp add: continuous_map_def)
    show "f ` topspace X = topspace Y" "g ` topspace Y = topspace X"
      using L by (force simp: continuous_map_closedin)+
    show "inj_on f (topspace X)" "inj_on g (topspace Y)"
      using L unfolding inj_on_def by metis+
  qed
next
  assume ?rhs
  then show ?lhs
    by (auto simp: homeomorphic_maps_def homeomorphic_imp_continuous_map)
qed

lemma homeomorphic_maps_imp_map:
    "homeomorphic_maps X Y f g \<Longrightarrow> homeomorphic_map X Y f"
  using homeomorphic_maps_map by blast

lemma homeomorphic_map_maps:
     "homeomorphic_map X Y f \<longleftrightarrow> (\<exists>g. homeomorphic_maps X Y f g)"
  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have L: "continuous_map X Y f" "open_map X Y f" "closed_map X Y f"
    "f ` (topspace X) = topspace Y" "inj_on f (topspace X)"
    by (auto simp: homeomorphic_eq_everything_map)
  have X: "\<And>x. x \<in> topspace X \<Longrightarrow> f x \<in> topspace Y \<and> inv_into (topspace X) f (f x) = x"
    using L by auto
  have Y: "\<And>y. y \<in> topspace Y \<Longrightarrow> inv_into (topspace X) f y \<in> topspace X \<and> f (inv_into (topspace X) f y) = y"
    by (simp add: L f_inv_into_f inv_into_into)
  have "homeomorphic_maps X Y f (inv_into (topspace X) f)"
    unfolding homeomorphic_maps_def
  proof (intro conjI L)
    show "continuous_map Y X (inv_into (topspace X) f)"
      by (simp add: L X Y flip: open_eq_continuous_inverse_map [where f=f])
  next
    show "\<forall>x\<in>topspace X. inv_into (topspace X) f (f x) = x"
         "\<forall>y\<in>topspace Y. f (inv_into (topspace X) f y) = y"
      using X Y by auto
  qed
  then show ?rhs
    by metis
next
  assume ?rhs
  then show ?lhs
    using homeomorphic_maps_map by blast
qed

lemma homeomorphic_maps_involution:
   "\<lbrakk>continuous_map X X f; \<And>x. x \<in> topspace X \<Longrightarrow> f(f x) = x\<rbrakk> \<Longrightarrow> homeomorphic_maps X X f f"
  by (auto simp: homeomorphic_maps_def)

lemma homeomorphic_map_involution:
   "\<lbrakk>continuous_map X X f; \<And>x. x \<in> topspace X \<Longrightarrow> f(f x) = x\<rbrakk> \<Longrightarrow> homeomorphic_map X X f"
  using homeomorphic_maps_involution homeomorphic_maps_map by blast

lemma homeomorphic_map_openness:
  assumes hom: "homeomorphic_map X Y f" and U: "U \<subseteq> topspace X"
  shows "openin Y (f ` U) \<longleftrightarrow> openin X U"
proof -
  obtain g where "homeomorphic_maps X Y f g"
    using assms by (auto simp: homeomorphic_map_maps)
  then have g: "homeomorphic_map Y X g" and gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g(f x) = x"
    by (auto simp: homeomorphic_maps_map)
  then have "openin X U \<Longrightarrow> openin Y (f ` U)"
    using hom homeomorphic_imp_open_map open_map_def by blast
  show "openin Y (f ` U) = openin X U"
  proof
    assume L: "openin Y (f ` U)"
    have "U = g ` (f ` U)"
      using U gf by force
    then show "openin X U"
      by (metis L homeomorphic_imp_open_map open_map_def g)
  next
    assume "openin X U"
    then show "openin Y (f ` U)"
      using hom homeomorphic_imp_open_map open_map_def by blast
  qed
qed


lemma homeomorphic_map_closedness:
  assumes hom: "homeomorphic_map X Y f" and U: "U \<subseteq> topspace X"
  shows "closedin Y (f ` U) \<longleftrightarrow> closedin X U"
proof -
  obtain g where "homeomorphic_maps X Y f g"
    using assms by (auto simp: homeomorphic_map_maps)
  then have g: "homeomorphic_map Y X g" and gf: "\<And>x. x \<in> topspace X \<Longrightarrow> g(f x) = x"
    by (auto simp: homeomorphic_maps_map)
  then have "closedin X U \<Longrightarrow> closedin Y (f ` U)"
    using hom homeomorphic_imp_closed_map closed_map_def by blast
  show "closedin Y (f ` U) = closedin X U"
  proof
    assume L: "closedin Y (f ` U)"
    have "U = g ` (f ` U)"
      using U gf by force
    then show "closedin X U"
      by (metis L homeomorphic_imp_closed_map closed_map_def g)
  next
    assume "closedin X U"
    then show "closedin Y (f ` U)"
      using hom homeomorphic_imp_closed_map closed_map_def by blast
  qed
qed

lemma homeomorphic_map_openness_eq:
     "homeomorphic_map X Y f \<Longrightarrow> openin X U \<longleftrightarrow> U \<subseteq> topspace X \<and> openin Y (f ` U)"
  by (meson homeomorphic_map_openness openin_closedin_eq)

lemma homeomorphic_map_closedness_eq:
    "homeomorphic_map X Y f \<Longrightarrow> closedin X U \<longleftrightarrow> U \<subseteq> topspace X \<and> closedin Y (f ` U)"
  by (meson closedin_subset homeomorphic_map_closedness)

lemma all_openin_homeomorphic_image:
  assumes "homeomorphic_map X Y f"
  shows "(\<forall>V. openin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. openin X U \<longrightarrow> P(f ` U))"  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (meson assms homeomorphic_map_openness_eq)
next
  assume ?rhs
  then show ?lhs
    by (metis (no_types, lifting) assms homeomorphic_imp_surjective_map homeomorphic_map_openness openin_subset subset_image_iff)
qed

lemma all_closedin_homeomorphic_image:
  assumes "homeomorphic_map X Y f"
  shows "(\<forall>V. closedin Y V \<longrightarrow> P V) \<longleftrightarrow> (\<forall>U. closedin X U \<longrightarrow> P(f ` U))"  (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (meson assms homeomorphic_map_closedness_eq)
next
  assume ?rhs
  then show ?lhs
    by (metis (no_types, lifting) assms homeomorphic_imp_surjective_map homeomorphic_map_closedness closedin_subset subset_image_iff)
qed


lemma homeomorphic_map_derived_set_of:
  assumes hom: "homeomorphic_map X Y f" and S: "S \<subseteq> topspace X"
  shows "Y derived_set_of (f ` S) = f ` (X derived_set_of S)"
proof -
  have fim: "f ` (topspace X) = topspace Y" and inj: "inj_on f (topspace X)"
    using hom by (auto simp: homeomorphic_eq_everything_map)
  have iff: "(\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)) =
            (\<forall>T. T \<subseteq> topspace Y \<longrightarrow> f x \<in> T \<longrightarrow> openin Y T \<longrightarrow> (\<exists>y. y \<noteq> f x \<and> y \<in> f ` S \<and> y \<in> T))"
    if "x \<in> topspace X" for x
  proof -
    have 1: "(x \<in> T \<and> openin X T) = (T \<subseteq> topspace X \<and> f x \<in> f ` T \<and> openin Y (f ` T))" for T
      by (meson hom homeomorphic_map_openness_eq inj inj_on_image_mem_iff that)
    have 2: "(\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T) = (\<exists>y. y \<noteq> f x \<and> y \<in> f ` S \<and> y \<in> f ` T)" (is "?lhs = ?rhs")
      if "T \<subseteq> topspace X \<and> f x \<in> f ` T \<and> openin Y (f ` T)" for T
    proof
      show "?lhs \<Longrightarrow> ?rhs"
        by (meson "1" imageI inj inj_on_eq_iff inj_on_subset that)
      show "?rhs \<Longrightarrow> ?lhs"
        using S inj inj_onD that by fastforce
    qed
    show ?thesis
      apply (simp flip: fim add: all_subset_image)
      apply (simp flip: imp_conjL)
      by (intro all_cong1 imp_cong 1 2)
  qed
  have *: "\<lbrakk>T = f ` S; \<And>x. x \<in> S \<Longrightarrow> P x \<longleftrightarrow> Q(f x)\<rbrakk> \<Longrightarrow> {y. y \<in> T \<and> Q y} = f ` {x \<in> S. P x}" for T S P Q
    by auto
  show ?thesis
    unfolding derived_set_of_def
    apply (rule *)
    using fim apply blast
    using iff openin_subset by force
qed


lemma homeomorphic_map_closure_of:
  assumes hom: "homeomorphic_map X Y f" and S: "S \<subseteq> topspace X"
  shows "Y closure_of (f ` S) = f ` (X closure_of S)"
  unfolding closure_of
  using homeomorphic_imp_surjective_map [OF hom] S
  by (auto simp: in_derived_set_of homeomorphic_map_derived_set_of [OF assms])

lemma homeomorphic_map_interior_of:
  assumes hom: "homeomorphic_map X Y f" and S: "S \<subseteq> topspace X"
  shows "Y interior_of (f ` S) = f ` (X interior_of S)"
proof -
  { fix y
    assume "y \<in> topspace Y" and "y \<notin> Y closure_of (topspace Y - f ` S)"
    then have "y \<in> f ` (topspace X - X closure_of (topspace X - S))"
      using homeomorphic_eq_everything_map [THEN iffD1, OF hom] homeomorphic_map_closure_of [OF hom]
      by (metis DiffI Diff_subset S closure_of_subset_topspace inj_on_image_set_diff) }
  moreover
  { fix x
    assume "x \<in> topspace X"
    then have "f x \<in> topspace Y"
      using hom homeomorphic_imp_surjective_map by blast }
  moreover
  { fix x
    assume "x \<in> topspace X" and "x \<notin> X closure_of (topspace X - S)" and "f x \<in> Y closure_of (topspace Y - f ` S)"
    then have "False"
      using homeomorphic_map_closure_of [OF hom] hom
      unfolding homeomorphic_eq_everything_map
      by (metis (no_types, lifting) Diff_subset S closure_of_subset_topspace inj_on_image_mem_iff_alt inj_on_image_set_diff) }
  ultimately  show ?thesis
    by (auto simp: interior_of_closure_of)
qed

lemma homeomorphic_map_frontier_of:
  assumes hom: "homeomorphic_map X Y f" and S: "S \<subseteq> topspace X"
  shows "Y frontier_of (f ` S) = f ` (X frontier_of S)"
  unfolding frontier_of_def
proof (intro equalityI subsetI DiffI)
  fix y
  assume "y \<in> Y closure_of f ` S - Y interior_of f ` S"
  then show "y \<in> f ` (X closure_of S - X interior_of S)"
    using S hom homeomorphic_map_closure_of homeomorphic_map_interior_of by fastforce
next
  fix y
  assume "y \<in> f ` (X closure_of S - X interior_of S)"
  then show "y \<in> Y closure_of f ` S"
    using S hom homeomorphic_map_closure_of by fastforce
next
  fix x
  assume "x \<in> f ` (X closure_of S - X interior_of S)"
  then obtain y where y: "x = f y" "y \<in> X closure_of S" "y \<notin> X interior_of S"
    by blast
  then have "y \<in> topspace X"
    by (simp add: in_closure_of)
  then have "f y \<notin> f ` (X interior_of S)"
    by (meson hom homeomorphic_eq_everything_map inj_on_image_mem_iff_alt interior_of_subset_topspace y(3))
  then show "x \<notin> Y interior_of f ` S"
    using S hom homeomorphic_map_interior_of y(1) by blast
qed

lemma homeomorphic_maps_subtopologies:
   "\<lbrakk>homeomorphic_maps X Y f g;  f ` (topspace X \<inter> S) = topspace Y \<inter> T\<rbrakk>
        \<Longrightarrow> homeomorphic_maps (subtopology X S) (subtopology Y T) f g"
  unfolding homeomorphic_maps_def
  by (force simp: continuous_map_from_subtopology topspace_subtopology continuous_map_in_subtopology)

lemma homeomorphic_maps_subtopologies_alt:
     "\<lbrakk>homeomorphic_maps X Y f g; f ` (topspace X \<inter> S) \<subseteq> T; g ` (topspace Y \<inter> T) \<subseteq> S\<rbrakk>
      \<Longrightarrow> homeomorphic_maps (subtopology X S) (subtopology Y T) f g"
  unfolding homeomorphic_maps_def
  by (force simp: continuous_map_from_subtopology topspace_subtopology continuous_map_in_subtopology)

lemma homeomorphic_map_subtopologies:
   "\<lbrakk>homeomorphic_map X Y f; f ` (topspace X \<inter> S) = topspace Y \<inter> T\<rbrakk>
        \<Longrightarrow> homeomorphic_map (subtopology X S) (subtopology Y T) f"
  by (meson homeomorphic_map_maps homeomorphic_maps_subtopologies)

lemma homeomorphic_map_subtopologies_alt:
   "\<lbrakk>homeomorphic_map X Y f;
     \<And>x. \<lbrakk>x \<in> topspace X; f x \<in> topspace Y\<rbrakk> \<Longrightarrow> f x \<in> T \<longleftrightarrow> x \<in> S\<rbrakk>
    \<Longrightarrow> homeomorphic_map (subtopology X S) (subtopology Y T) f"
  unfolding homeomorphic_map_maps
  apply (erule ex_forward)
  apply (rule homeomorphic_maps_subtopologies)
  apply (auto simp: homeomorphic_maps_def continuous_map_def)
  by (metis IntI image_iff)


subsection\<open>Relation of homeomorphism between topological spaces\<close>

definition homeomorphic_space (infixr "homeomorphic'_space" 50)
  where "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"

lemma homeomorphic_space_refl: "X homeomorphic_space X"
  by (meson homeomorphic_maps_id homeomorphic_space_def)

lemma homeomorphic_space_sym:
   "X homeomorphic_space Y \<longleftrightarrow> Y homeomorphic_space X"
  unfolding homeomorphic_space_def by (metis homeomorphic_maps_sym)

lemma homeomorphic_space_trans:
     "\<lbrakk>X1 homeomorphic_space X2; X2 homeomorphic_space X3\<rbrakk> \<Longrightarrow> X1 homeomorphic_space X3"
  unfolding homeomorphic_space_def by (metis homeomorphic_maps_compose)

lemma homeomorphic_space:
     "X homeomorphic_space Y \<longleftrightarrow> (\<exists>f. homeomorphic_map X Y f)"
  by (simp add: homeomorphic_map_maps homeomorphic_space_def)

lemma homeomorphic_maps_imp_homeomorphic_space:
     "homeomorphic_maps X Y f g \<Longrightarrow> X homeomorphic_space Y"
  unfolding homeomorphic_space_def by metis

lemma homeomorphic_map_imp_homeomorphic_space:
     "homeomorphic_map X Y f \<Longrightarrow> X homeomorphic_space Y"
  unfolding homeomorphic_map_maps
  using homeomorphic_space_def by blast

lemma homeomorphic_empty_space:
     "X homeomorphic_space Y \<Longrightarrow> topspace X = {} \<longleftrightarrow> topspace Y = {}"
  by (metis homeomorphic_imp_surjective_map homeomorphic_space image_is_empty)

lemma homeomorphic_empty_space_eq:
  assumes "topspace X = {}"
    shows "X homeomorphic_space Y \<longleftrightarrow> topspace Y = {}"
proof -
  have "\<forall>f t. continuous_map X (t::'b topology) f"
    using assms continuous_map_on_empty by blast
  then show ?thesis
    by (metis (no_types) assms continuous_map_on_empty empty_iff homeomorphic_empty_space homeomorphic_maps_def homeomorphic_space_def)
qed

subsection\<open>Connected topological spaces\<close>

definition connected_space :: "'a topology \<Rightarrow> bool" where
  "connected_space X \<equiv>
        \<not>(\<exists>E1 E2. openin X E1 \<and> openin X E2 \<and>
                  topspace X \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"

definition connectedin :: "'a topology \<Rightarrow> 'a set \<Rightarrow> bool" where
  "connectedin X S \<equiv> S \<subseteq> topspace X \<and> connected_space (subtopology X S)"

lemma connectedin_subset_topspace: "connectedin X S \<Longrightarrow> S \<subseteq> topspace X"
  by (simp add: connectedin_def)

lemma connectedin_topspace:
     "connectedin X (topspace X) \<longleftrightarrow> connected_space X"
  by (simp add: connectedin_def)

lemma connected_space_subtopology:
     "connectedin X S \<Longrightarrow> connected_space (subtopology X S)"
  by (simp add: connectedin_def)

lemma connectedin_subtopology:
     "connectedin (subtopology X S) T \<longleftrightarrow> connectedin X T \<and> T \<subseteq> S"
  by (force simp: connectedin_def subtopology_subtopology topspace_subtopology inf_absorb2)

lemma connected_space_eq:
     "connected_space X \<longleftrightarrow>
      (\<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
  unfolding connected_space_def
  by (metis openin_Un openin_subset subset_antisym)

lemma connected_space_closedin:
     "connected_space X \<longleftrightarrow>
      (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> topspace X \<subseteq> E1 \<union> E2 \<and>
               E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})" (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have L: "\<And>E1 E2. \<lbrakk>openin X E1; E1 \<inter> E2 = {}; topspace X \<subseteq> E1 \<union> E2; openin X E2\<rbrakk> \<Longrightarrow> E1 = {} \<or> E2 = {}"
    by (simp add: connected_space_def)
  show ?rhs
    unfolding connected_space_def
  proof clarify
    fix E1 E2
    assume "closedin X E1" and "closedin X E2" and "topspace X \<subseteq> E1 \<union> E2" and "E1 \<inter> E2 = {}"
      and "E1 \<noteq> {}" and "E2 \<noteq> {}"
    have "E1 \<union> E2 = topspace X"
      by (meson Un_subset_iff \<open>closedin X E1\<close> \<open>closedin X E2\<close> \<open>topspace X \<subseteq> E1 \<union> E2\<close> closedin_def subset_antisym)
    then have "topspace X - E2 = E1"
      using \<open>E1 \<inter> E2 = {}\<close> by fastforce
    then have "topspace X = E1"
      using \<open>E1 \<noteq> {}\<close> L \<open>closedin X E1\<close> \<open>closedin X E2\<close> by blast
    then show "False"
      using \<open>E1 \<inter> E2 = {}\<close> \<open>E1 \<union> E2 = topspace X\<close> \<open>E2 \<noteq> {}\<close> by blast
  qed
next
  assume R: ?rhs
  show ?lhs
    unfolding connected_space_def
  proof clarify
    fix E1 E2
    assume "openin X E1" and "openin X E2" and "topspace X \<subseteq> E1 \<union> E2" and "E1 \<inter> E2 = {}"
      and "E1 \<noteq> {}" and "E2 \<noteq> {}"
    have "E1 \<union> E2 = topspace X"
      by (meson Un_subset_iff \<open>openin X E1\<close> \<open>openin X E2\<close> \<open>topspace X \<subseteq> E1 \<union> E2\<close> openin_closedin_eq subset_antisym)
    then have "topspace X - E2 = E1"
      using \<open>E1 \<inter> E2 = {}\<close> by fastforce
    then have "topspace X = E1"
      using \<open>E1 \<noteq> {}\<close> R \<open>openin X E1\<close> \<open>openin X E2\<close> by blast
    then show "False"
      using \<open>E1 \<inter> E2 = {}\<close> \<open>E1 \<union> E2 = topspace X\<close> \<open>E2 \<noteq> {}\<close> by blast
  qed
qed

lemma connected_space_closedin_eq:
     "connected_space X \<longleftrightarrow>
       (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and>
                E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
  apply (simp add: connected_space_closedin)
  apply (intro all_cong)
  using closedin_subset apply blast
  done

lemma connected_space_clopen_in:
     "connected_space X \<longleftrightarrow>
        (\<forall>T. openin X T \<and> closedin X T \<longrightarrow> T = {} \<or> T = topspace X)"
proof -
  have eq: "openin X E1 \<and> openin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> P
        \<longleftrightarrow> E2 = topspace X - E1 \<and> openin X E1 \<and> openin X E2 \<and> P" for E1 E2 P
    using openin_subset by blast
  show ?thesis
    unfolding connected_space_eq eq closedin_def
    by (auto simp: openin_closedin_eq)
qed

lemma connectedin:
     "connectedin X S \<longleftrightarrow>
        S \<subseteq> topspace X \<and>
         (\<nexists>E1 E2.
             openin X E1 \<and> openin X E2 \<and>
             S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 \<inter> S = {} \<and> E1 \<inter> S \<noteq> {} \<and> E2 \<inter> S \<noteq> {})"
proof -
  have *: "(\<exists>E1:: 'a set. \<exists>E2:: 'a set. (\<exists>T1:: 'a set. P1 T1 \<and> E1 = f1 T1) \<and> (\<exists>T2:: 'a set. P2 T2 \<and> E2 = f2 T2) \<and>
             R E1 E2) \<longleftrightarrow> (\<exists>T1 T2. P1 T1 \<and> P2 T2 \<and> R(f1 T1) (f2 T2))" for P1 f1 P2 f2 R
    by auto
  show ?thesis
    unfolding connectedin_def connected_space_def openin_subtopology topspace_subtopology Not_eq_iff *
    apply (intro conj_cong arg_cong [where f=Not] ex_cong1 refl)
    apply (blast elim: dest!: openin_subset)+
    done
qed

lemma connectedin_iff_connected_real [simp]:
     "connectedin euclideanreal S \<longleftrightarrow> connected S"
    by (simp add: connected_def connectedin)

lemma connectedin_closedin:
   "connectedin X S \<longleftrightarrow>
        S \<subseteq> topspace X \<and>
        \<not>(\<exists>E1 E2. closedin X E1 \<and> closedin X E2 \<and>
                  S \<subseteq> (E1 \<union> E2) \<and>
                  (E1 \<inter> E2 \<inter> S = {}) \<and>
                  \<not>(E1 \<inter> S = {}) \<and> \<not>(E2 \<inter> S = {}))"
proof -
  have *: "(\<exists>E1:: 'a set. \<exists>E2:: 'a set. (\<exists>T1:: 'a set. P1 T1 \<and> E1 = f1 T1) \<and> (\<exists>T2:: 'a set. P2 T2 \<and> E2 = f2 T2) \<and>
             R E1 E2) \<longleftrightarrow> (\<exists>T1 T2. P1 T1 \<and> P2 T2 \<and> R(f1 T1) (f2 T2))" for P1 f1 P2 f2 R
    by auto
  show ?thesis
    unfolding connectedin_def connected_space_closedin closedin_subtopology topspace_subtopology Not_eq_iff *
    apply (intro conj_cong arg_cong [where f=Not] ex_cong1 refl)
    apply (blast elim: dest!: openin_subset)+
    done
qed

lemma connectedin_empty [simp]: "connectedin X {}"
  by (simp add: connectedin)

lemma connected_space_topspace_empty:
     "topspace X = {} \<Longrightarrow> connected_space X"
  using connectedin_topspace by fastforce

lemma connectedin_sing [simp]: "connectedin X {a} \<longleftrightarrow> a \<in> topspace X"
  by (simp add: connectedin)

lemma connectedin_absolute [simp]:
  "connectedin (subtopology X S) S \<longleftrightarrow> connectedin X S"
  apply (simp only: connectedin_def topspace_subtopology subtopology_subtopology)
  apply (intro conj_cong imp_cong arg_cong [where f=Not] all_cong1 ex_cong1 refl)
  by auto

lemma connectedin_Union:
  assumes \<U>: "\<And>S. S \<in> \<U> \<Longrightarrow> connectedin X S" and ne: "\<Inter>\<U> \<noteq> {}"
  shows "connectedin X (\<Union>\<U>)"
proof -
  have "\<Union>\<U> \<subseteq> topspace X"
    using \<U> by (simp add: Union_least connectedin_def)
  moreover have False
    if "openin X E1" "openin X E2" and cover: "\<Union>\<U> \<subseteq> E1 \<union> E2" and disj: "E1 \<inter> E2 \<inter> \<Union>\<U> = {}"
       and overlap1: "E1 \<inter> \<Union>\<U> \<noteq> {}" and overlap2: "E2 \<inter> \<Union>\<U> \<noteq> {}"
      for E1 E2
  proof -
    have disjS: "E1 \<inter> E2 \<inter> S = {}" if "S \<in> \<U>" for S
      using Diff_triv that disj by auto
    have coverS: "S \<subseteq> E1 \<union> E2" if "S \<in> \<U>" for S
      using that cover by blast
    have "\<U> \<noteq> {}"
      using overlap1 by blast
    obtain a where a: "\<And>U. U \<in> \<U> \<Longrightarrow> a \<in> U"
      using ne by force
    with \<open>\<U> \<noteq> {}\<close> have "a \<in> \<Union>\<U>"
      by blast
    then consider "a \<in> E1" | "a \<in> E2"
      using \<open>\<Union>\<U> \<subseteq> E1 \<union> E2\<close> by auto
    then show False
    proof cases
      case 1
      then obtain b S where "b \<in> E2" "b \<in> S" "S \<in> \<U>"
        using overlap2 by blast
      then show ?thesis
        using "1" \<open>openin X E1\<close> \<open>openin X E2\<close> disjS coverS a [OF \<open>S \<in> \<U>\<close>]  \<U>[OF \<open>S \<in> \<U>\<close>]
        unfolding connectedin
        by (meson disjoint_iff_not_equal)
    next
      case 2
      then obtain b S where "b \<in> E1" "b \<in> S" "S \<in> \<U>"
        using overlap1 by blast
      then show ?thesis
        using "2" \<open>openin X E1\<close> \<open>openin X E2\<close> disjS coverS a [OF \<open>S \<in> \<U>\<close>]  \<U>[OF \<open>S \<in> \<U>\<close>]
        unfolding connectedin
        by (meson disjoint_iff_not_equal)
    qed
  qed
  ultimately show ?thesis
    unfolding connectedin by blast
qed

lemma connectedin_Un:
     "\<lbrakk>connectedin X S; connectedin X T; S \<inter> T \<noteq> {}\<rbrakk> \<Longrightarrow> connectedin X (S \<union> T)"
  using connectedin_Union [of "{S,T}"] by auto

lemma connected_space_subconnected:
  "connected_space X \<longleftrightarrow> (\<forall>x \<in> topspace X. \<forall>y \<in> topspace X. \<exists>S. connectedin X S \<and> x \<in> S \<and> y \<in> S)" (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    using connectedin_topspace by blast
next
  assume R [rule_format]: ?rhs
  have False if "openin X U" "openin X V" and disj: "U \<inter> V = {}" and cover: "topspace X \<subseteq> U \<union> V"
    and "U \<noteq> {}" "V \<noteq> {}" for U V
  proof -
    obtain u v where "u \<in> U" "v \<in> V"
      using \<open>U \<noteq> {}\<close> \<open>V \<noteq> {}\<close> by auto
    then obtain T where "u \<in> T" "v \<in> T" and T: "connectedin X T"
      using R [of u v] that
      by (meson \<open>openin X U\<close> \<open>openin X V\<close> subsetD openin_subset)
    then show False
      using that unfolding connectedin
      by (metis IntI \<open>u \<in> U\<close> \<open>v \<in> V\<close> empty_iff inf_bot_left subset_trans)
  qed
  then show ?lhs
    by (auto simp: connected_space_def)
qed

lemma connectedin_intermediate_closure_of:
  assumes "connectedin X S" "S \<subseteq> T" "T \<subseteq> X closure_of S"
  shows "connectedin X T"
proof -
  have S: "S \<subseteq> topspace X"and T: "T \<subseteq> topspace X"
    using assms by (meson closure_of_subset_topspace dual_order.trans)+
  show ?thesis
  using assms
  apply (simp add: connectedin closure_of_subset_topspace S T)
  apply (elim all_forward imp_forward2 asm_rl)
  apply (blast dest: openin_Int_closure_of_eq_empty [of X _ S])+
  done
qed

lemma connectedin_closure_of:
     "connectedin X S \<Longrightarrow> connectedin X (X closure_of S)"
  by (meson closure_of_subset connectedin_def connectedin_intermediate_closure_of subset_refl)

lemma connectedin_separation:
  "connectedin X S \<longleftrightarrow>
        S \<subseteq> topspace X \<and>
        (\<nexists>C1 C2. C1 \<union> C2 = S \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> C1 \<inter> X closure_of C2 = {} \<and> C2 \<inter> X closure_of C1 = {})" (is "?lhs = ?rhs")
  unfolding connectedin_def connected_space_closedin_eq closedin_Int_closure_of topspace_subtopology
  apply (intro conj_cong refl arg_cong [where f=Not])
  apply (intro ex_cong1 iffI, blast)
  using closure_of_subset_Int by force

lemma connectedin_eq_not_separated:
   "connectedin X S \<longleftrightarrow>
         S \<subseteq> topspace X \<and>
         (\<nexists>C1 C2. C1 \<union> C2 = S \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2)"
  apply (simp add: separatedin_def connectedin_separation)
  apply (intro conj_cong all_cong1 refl, blast)
  done

lemma connectedin_eq_not_separated_subset:
  "connectedin X S \<longleftrightarrow>
      S \<subseteq> topspace X \<and> (\<nexists>C1 C2. S \<subseteq> C1 \<union> C2 \<and> S \<inter> C1 \<noteq> {} \<and> S \<inter> C2 \<noteq> {} \<and> separatedin X C1 C2)"
proof -
  have *: "\<forall>C1 C2. S \<subseteq> C1 \<union> C2 \<longrightarrow> S \<inter> C1 = {} \<or> S \<inter> C2 = {} \<or> \<not> separatedin X C1 C2"
    if "\<And>C1 C2. C1 \<union> C2 = S \<longrightarrow> C1 = {} \<or> C2 = {} \<or> \<not> separatedin X C1 C2"
  proof (intro allI)
    fix C1 C2
    show "S \<subseteq> C1 \<union> C2 \<longrightarrow> S \<inter> C1 = {} \<or> S \<inter> C2 = {} \<or> \<not> separatedin X C1 C2"
      using that [of "S \<inter> C1" "S \<inter> C2"]
      by (auto simp: separatedin_mono)
  qed
  show ?thesis
    apply (simp add: connectedin_eq_not_separated)
    apply (intro conj_cong refl iffI *)
    apply (blast elim!: all_forward)+
    done
qed

lemma connected_space_eq_not_separated:
     "connected_space X \<longleftrightarrow>
      (\<nexists>C1 C2. C1 \<union> C2 = topspace X \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2)"
  by (simp add: connectedin_eq_not_separated flip: connectedin_topspace)

lemma connected_space_eq_not_separated_subset:
  "connected_space X \<longleftrightarrow>
    (\<nexists>C1 C2. topspace X \<subseteq> C1 \<union> C2 \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2)"
  apply (simp add: connected_space_eq_not_separated)
  apply (intro all_cong1)
  by (metis Un_absorb dual_order.antisym separatedin_def subset_refl sup_mono)

lemma connectedin_subset_separated_union:
     "\<lbrakk>connectedin X C; separatedin X S T; C \<subseteq> S \<union> T\<rbrakk> \<Longrightarrow> C \<subseteq> S \<or> C \<subseteq> T"
  unfolding connectedin_eq_not_separated_subset  by blast

lemma connectedin_nonseparated_union:
   "\<lbrakk>connectedin X S; connectedin X T; \<not>separatedin X S T\<rbrakk> \<Longrightarrow> connectedin X (S \<union> T)"
  apply (simp add: connectedin_eq_not_separated_subset, auto)
    apply (metis (no_types, hide_lams) Diff_subset_conv Diff_triv disjoint_iff_not_equal separatedin_mono sup_commute)
  apply (metis (no_types, hide_lams) Diff_subset_conv Diff_triv disjoint_iff_not_equal separatedin_mono separatedin_sym sup_commute)
  by (meson disjoint_iff_not_equal)

lemma connected_space_closures:
     "connected_space X \<longleftrightarrow>
        (\<nexists>e1 e2. e1 \<union> e2 = topspace X \<and> X closure_of e1 \<inter> X closure_of e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding connected_space_closedin_eq
    by (metis Un_upper1 Un_upper2 closedin_closure_of closure_of_Un closure_of_eq_empty closure_of_topspace)
next
  assume ?rhs
  then show ?lhs
    unfolding connected_space_closedin_eq
    by (metis closure_of_eq)
qed

lemma connectedin_inter_frontier_of:
  assumes "connectedin X S" "S \<inter> T \<noteq> {}" "S - T \<noteq> {}"
  shows "S \<inter> X frontier_of T \<noteq> {}"
proof -
  have "S \<subseteq> topspace X" and *:
    "\<And>E1 E2. openin X E1 \<longrightarrow> openin X E2 \<longrightarrow> E1 \<inter> E2 \<inter> S = {} \<longrightarrow> S \<subseteq> E1 \<union> E2 \<longrightarrow> E1 \<inter> S = {} \<or> E2 \<inter> S = {}"
    using \<open>connectedin X S\<close> by (auto simp: connectedin)
  have "S - (topspace X \<inter> T) \<noteq> {}"
    using assms(3) by blast
  moreover
  have "S \<inter> topspace X \<inter> T \<noteq> {}"
    using assms(1) assms(2) connectedin by fastforce
  moreover
  have False if "S \<inter> T \<noteq> {}" "S - T \<noteq> {}" "T \<subseteq> topspace X" "S \<inter> X frontier_of T = {}" for T
  proof -
    have null: "S \<inter> (X closure_of T - X interior_of T) = {}"
      using that unfolding frontier_of_def by blast
    have 1: "X interior_of T \<inter> (topspace X - X closure_of T) \<inter> S = {}"
      by (metis Diff_disjoint inf_bot_left interior_of_Int interior_of_complement interior_of_empty)
    have 2: "S \<subseteq> X interior_of T \<union> (topspace X - X closure_of T)"
      using that \<open>S \<subseteq> topspace X\<close> null by auto
    have 3: "S \<inter> X interior_of T \<noteq> {}"
      using closure_of_subset that(1) that(3) null by fastforce
    show ?thesis
      using null \<open>S \<subseteq> topspace X\<close> that * [of "X interior_of T" "topspace X - X closure_of T"]
      apply (clarsimp simp add: openin_diff 1 2)
      apply (simp add: Int_commute Diff_Int_distrib 3)
      by (metis Int_absorb2 contra_subsetD interior_of_subset)
  qed
  ultimately show ?thesis
    by (metis Int_lower1 frontier_of_restrict inf_assoc)
qed

lemma connectedin_continuous_map_image:
  assumes f: "continuous_map X Y f" and "connectedin X S"
  shows "connectedin Y (f ` S)"
proof -
  have "S \<subseteq> topspace X" and *:
    "\<And>E1 E2. openin X E1 \<longrightarrow> openin X E2 \<longrightarrow> E1 \<inter> E2 \<inter> S = {} \<longrightarrow> S \<subseteq> E1 \<union> E2 \<longrightarrow> E1 \<inter> S = {} \<or> E2 \<inter> S = {}"
    using \<open>connectedin X S\<close> by (auto simp: connectedin)
  show ?thesis
    unfolding connectedin connected_space_def
  proof (intro conjI notI; clarify)
    show "f x \<in> topspace Y" if  "x \<in> S" for x
      using \<open>S \<subseteq> topspace X\<close> continuous_map_image_subset_topspace f that by blast
  next
    fix U V
    let ?U = "{x \<in> topspace X. f x \<in> U}"
    let ?V = "{x \<in> topspace X. f x \<in> V}"
    assume UV: "openin Y U" "openin Y V" "f ` S \<subseteq> U \<union> V" "U \<inter> V \<inter> f ` S = {}" "U \<inter> f ` S \<noteq> {}" "V \<inter> f ` S \<noteq> {}"
    then have 1: "?U \<inter> ?V \<inter> S = {}"
      by auto
    have 2: "openin X ?U" "openin X ?V"
      using \<open>openin Y U\<close> \<open>openin Y V\<close> continuous_map f by fastforce+
    show "False"
      using  * [of ?U ?V] UV \<open>S \<subseteq> topspace X\<close>
      by (auto simp: 1 2)
  qed
qed

lemma homeomorphic_connected_space:
     "X homeomorphic_space Y \<Longrightarrow> connected_space X \<longleftrightarrow> connected_space Y"
  unfolding homeomorphic_space_def homeomorphic_maps_def
  apply safe
  apply (metis connectedin_continuous_map_image connected_space_subconnected continuous_map_image_subset_topspace image_eqI image_subset_iff)
  by (metis (no_types, hide_lams) connectedin_continuous_map_image connectedin_topspace continuous_map_def continuous_map_image_subset_topspace imageI set_eq_subset subsetI)

lemma homeomorphic_map_connectedness:
  assumes f: "homeomorphic_map X Y f" and U: "U \<subseteq> topspace X"
  shows "connectedin Y (f ` U) \<longleftrightarrow> connectedin X U"
proof -
  have 1: "f ` U \<subseteq> topspace Y \<longleftrightarrow> U \<subseteq> topspace X"
    using U f homeomorphic_imp_surjective_map by blast
  moreover have "connected_space (subtopology Y (f ` U)) \<longleftrightarrow> connected_space (subtopology X U)"
  proof (rule homeomorphic_connected_space)
    have "f ` U \<subseteq> topspace Y"
      by (simp add: U 1)
    then have "topspace Y \<inter> f ` U = f ` U"
      by (simp add: subset_antisym)
    then show "subtopology Y (f ` U) homeomorphic_space subtopology X U"
      by (metis (no_types) Int_subset_iff U f homeomorphic_map_imp_homeomorphic_space homeomorphic_map_subtopologies homeomorphic_space_sym subset_antisym subset_refl)
  qed
  ultimately show ?thesis
    by (auto simp: connectedin_def)
qed

lemma homeomorphic_map_connectedness_eq:
   "homeomorphic_map X Y f
        \<Longrightarrow> connectedin X U \<longleftrightarrow>
             U \<subseteq> topspace X \<and> connectedin Y (f ` U)"
  using homeomorphic_map_connectedness connectedin_subset_topspace by metis

lemma connectedin_discrete_topology:
   "connectedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a})"
proof (cases "S \<subseteq> U")
  case True
  show ?thesis
  proof (cases "S = {}")
    case False
    moreover have "connectedin (discrete_topology U) S \<longleftrightarrow> (\<exists>a. S = {a})"
      apply safe
      using False connectedin_inter_frontier_of insert_Diff apply fastforce
      using True by auto
    ultimately show ?thesis
      by auto
  qed simp
next
  case False
  then show ?thesis
    by (simp add: connectedin_def)
qed

lemma connected_space_discrete_topology:
     "connected_space (discrete_topology U) \<longleftrightarrow> (\<exists>a. U \<subseteq> {a})"
  by (metis connectedin_discrete_topology connectedin_topspace order_refl topspace_discrete_topology)


subsection\<open>Compact sets\<close>

definition compactin where
 "compactin X S \<longleftrightarrow>
     S \<subseteq> topspace X \<and>
     (\<forall>\<U>. (\<forall>U \<in> \<U>. openin X U) \<and> S \<subseteq> \<Union>\<U>
          \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union>\<F>))"

definition compact_space where
   "compact_space X \<equiv> compactin X (topspace X)"

lemma compact_space_alt:
   "compact_space X \<longleftrightarrow>
        (\<forall>\<U>. (\<forall>U \<in> \<U>. openin X U) \<and> topspace X \<subseteq> \<Union>\<U>
            \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union>\<F>))"
  by (simp add: compact_space_def compactin_def)

lemma compact_space:
   "compact_space X \<longleftrightarrow>
        (\<forall>\<U>. (\<forall>U \<in> \<U>. openin X U) \<and> \<Union>\<U> = topspace X
            \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> \<Union>\<F> = topspace X))"
  unfolding compact_space_alt
  using openin_subset by fastforce

lemma compactin_euclideanreal_iff [simp]: "compactin euclideanreal S \<longleftrightarrow> compact S"
  by (simp add: compact_eq_Heine_Borel compactin_def) meson

lemma compactin_absolute [simp]:
   "compactin (subtopology X S) S \<longleftrightarrow> compactin X S"
proof -
  have eq: "(\<forall>U \<in> \<U>. \<exists>Y. openin X Y \<and> U = Y \<inter> S) \<longleftrightarrow> \<U> \<subseteq> (\<lambda>Y. Y \<inter> S) ` {y. openin X y}" for \<U>
    by auto
  show ?thesis
    by (auto simp: compactin_def topspace_subtopology openin_subtopology eq imp_conjL all_subset_image exists_finite_subset_image)
qed

lemma compactin_subspace: "compactin X S \<longleftrightarrow> S \<subseteq> topspace X \<and> compact_space (subtopology X S)"
  unfolding compact_space_def topspace_subtopology
  by (metis compactin_absolute compactin_def inf.absorb2)

lemma compact_space_subtopology: "compactin X S \<Longrightarrow> compact_space (subtopology X S)"
  by (simp add: compactin_subspace)

lemma compactin_subtopology: "compactin (subtopology X S) T \<longleftrightarrow> compactin X T \<and> T \<subseteq> S"
apply (simp add: compactin_subspace topspace_subtopology)
  by (metis inf.orderE inf_commute subtopology_subtopology)


lemma compactin_subset_topspace: "compactin X S \<Longrightarrow> S \<subseteq> topspace X"
  by (simp add: compactin_subspace)

lemma compactin_contractive:
   "\<lbrakk>compactin X' S; topspace X' = topspace X;
     \<And>U. openin X U \<Longrightarrow> openin X' U\<rbrakk> \<Longrightarrow> compactin X S"
  by (simp add: compactin_def)

lemma finite_imp_compactin:
   "\<lbrakk>S \<subseteq> topspace X; finite S\<rbrakk> \<Longrightarrow> compactin X S"
  by (metis compactin_subspace compact_space finite_UnionD inf.absorb_iff2 order_refl topspace_subtopology)

lemma compactin_empty [iff]: "compactin X {}"
  by (simp add: finite_imp_compactin)

lemma compact_space_topspace_empty:
   "topspace X = {} \<Longrightarrow> compact_space X"
  by (simp add: compact_space_def)

lemma finite_imp_compactin_eq:
   "finite S \<Longrightarrow> (compactin X S \<longleftrightarrow> S \<subseteq> topspace X)"
  using compactin_subset_topspace finite_imp_compactin by blast

lemma compactin_sing [simp]: "compactin X {a} \<longleftrightarrow> a \<in> topspace X"
  by (simp add: finite_imp_compactin_eq)

lemma closed_compactin:
  assumes XK: "compactin X K" and "C \<subseteq> K" and XC: "closedin X C"
  shows "compactin X C"
  unfolding compactin_def
proof (intro conjI allI impI)
  show "C \<subseteq> topspace X"
    by (simp add: XC closedin_subset)
next
  fix \<U> :: "'a set set"
  assume \<U>: "Ball \<U> (openin X) \<and> C \<subseteq> \<Union>\<U>"
  have "(\<forall>U\<in>insert (topspace X - C) \<U>. openin X U)"
    using XC \<U> by blast
  moreover have "K \<subseteq> \<Union>insert (topspace X - C) \<U>"
    using \<U> XK compactin_subset_topspace by fastforce
  ultimately obtain \<F> where "finite \<F>" "\<F> \<subseteq> insert (topspace X - C) \<U>" "K \<subseteq> \<Union>\<F>"
    using assms unfolding compactin_def by metis
  moreover have "openin X (topspace X - C)"
    using XC by auto
  ultimately show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> C \<subseteq> \<Union>\<F>"
    using \<open>C \<subseteq> K\<close>
    by (rule_tac x="\<F> - {topspace X - C}" in exI) auto
qed

lemma closedin_compact_space:
   "\<lbrakk>compact_space X; closedin X S\<rbrakk> \<Longrightarrow> compactin X S"
  by (simp add: closed_compactin closedin_subset compact_space_def)

lemma compact_Int_closedin:
  assumes "compactin X S" "closedin X T" shows "compactin X (S \<inter> T)"
proof -
  have "compactin (subtopology X S) (S \<inter> T)"
    by (metis assms closedin_compact_space closedin_subtopology compactin_subspace inf_commute)
  then show ?thesis
    by (simp add: compactin_subtopology)
qed

lemma closed_Int_compactin: "\<lbrakk>closedin X S; compactin X T\<rbrakk> \<Longrightarrow> compactin X (S \<inter> T)"
  by (metis compact_Int_closedin inf_commute)

lemma compactin_Un:
  assumes S: "compactin X S" and T: "compactin X T" shows "compactin X (S \<union> T)"
  unfolding compactin_def
proof (intro conjI allI impI)
  show "S \<union> T \<subseteq> topspace X"
    using assms by (auto simp: compactin_def)
next
  fix \<U> :: "'a set set"
  assume \<U>: "Ball \<U> (openin X) \<and> S \<union> T \<subseteq> \<Union>\<U>"
  with S obtain \<F> where \<V>: "finite \<F>" "\<F> \<subseteq> \<U>" "S \<subseteq> \<Union>\<F>"
    unfolding compactin_def by (meson sup.bounded_iff)
  obtain \<W> where "finite \<W>" "\<W> \<subseteq> \<U>" "T \<subseteq> \<Union>\<W>"
    using \<U> T
    unfolding compactin_def by (meson sup.bounded_iff)
  with \<V> show "\<exists>\<V>. finite \<V> \<and> \<V> \<subseteq> \<U> \<and> S \<union> T \<subseteq> \<Union>\<V>"
    by (rule_tac x="\<F> \<union> \<W>" in exI) auto
qed

lemma compactin_Union:
   "\<lbrakk>finite \<F>; \<And>S. S \<in> \<F> \<Longrightarrow> compactin X S\<rbrakk> \<Longrightarrow> compactin X (\<Union>\<F>)"
by (induction rule: finite_induct) (simp_all add: compactin_Un)

lemma compactin_subtopology_imp_compact:
  assumes "compactin (subtopology X S) K" shows "compactin X K"
  using assms
proof (clarsimp simp add: compactin_def topspace_subtopology)
  fix \<U>
  define \<V> where "\<V> \<equiv> (\<lambda>U. U \<inter> S) ` \<U>"
  assume "K \<subseteq> topspace X" and "K \<subseteq> S" and "\<forall>x\<in>\<U>. openin X x" and "K \<subseteq> \<Union>\<U>"
  then have "\<forall>V \<in> \<V>. openin (subtopology X S) V" "K \<subseteq> \<Union>\<V>"
    unfolding \<V>_def by (auto simp: openin_subtopology)
  moreover
  assume "\<forall>\<U>. (\<forall>x\<in>\<U>. openin (subtopology X S) x) \<and> K \<subseteq> \<Union>\<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> K \<subseteq> \<Union>\<F>)"
  ultimately obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<V>" "K \<subseteq> \<Union>\<F>"
    by meson
  then have \<F>: "\<exists>U. U \<in> \<U> \<and> V = U \<inter> S" if "V \<in> \<F>" for V
    unfolding \<V>_def using that by blast
  let ?\<F> = "(\<lambda>F. @U. U \<in> \<U> \<and> F = U \<inter> S) ` \<F>"
  show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> K \<subseteq> \<Union>\<F>"
  proof (intro exI conjI)
    show "finite ?\<F>"
      using \<open>finite \<F>\<close> by blast
    show "?\<F> \<subseteq> \<U>"
      using someI_ex [OF \<F>] by blast
    show "K \<subseteq> \<Union>?\<F>"
    proof clarsimp
      fix x
      assume "x \<in> K"
      then show "\<exists>V \<in> \<F>. x \<in> (SOME U. U \<in> \<U> \<and> V = U \<inter> S)"
        using \<open>K \<subseteq> \<Union>\<F>\<close> someI_ex [OF \<F>]
        by (metis (no_types, lifting) IntD1 Union_iff subsetCE)
    qed
  qed
qed

lemma compact_imp_compactin_subtopology:
  assumes "compactin X K" "K \<subseteq> S" shows "compactin (subtopology X S) K"
  using assms
proof (clarsimp simp add: compactin_def topspace_subtopology)
  fix \<U> :: "'a set set"
  define \<V> where "\<V> \<equiv> {V. openin X V \<and> (\<exists>U \<in> \<U>. U = V \<inter> S)}"
  assume "K \<subseteq> S" and "K \<subseteq> topspace X" and "\<forall>U\<in>\<U>. openin (subtopology X S) U" and "K \<subseteq> \<Union>\<U>"
  then have "\<forall>V \<in> \<V>. openin X V" "K \<subseteq> \<Union>\<V>"
    unfolding \<V>_def by (fastforce simp: subset_eq openin_subtopology)+
  moreover
  assume "\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> K \<subseteq> \<Union>\<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> K \<subseteq> \<Union>\<F>)"
  ultimately obtain \<F> where "finite \<F>" "\<F> \<subseteq> \<V>" "K \<subseteq> \<Union>\<F>"
    by meson
  let ?\<F> = "(\<lambda>F. F \<inter> S) ` \<F>"
  show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> K \<subseteq> \<Union>\<F>"
  proof (intro exI conjI)
    show "finite ?\<F>"
      using \<open>finite \<F>\<close> by blast
    show "?\<F> \<subseteq> \<U>"
      using \<V>_def \<open>\<F> \<subseteq> \<V>\<close> by blast
    show "K \<subseteq> \<Union>?\<F>"
      using \<open>K \<subseteq> \<Union>\<F>\<close> assms(2) by auto
  qed
qed


proposition compact_space_fip:
   "compact_space X \<longleftrightarrow>
    (\<forall>\<U>. (\<forall>C\<in>\<U>. closedin X C) \<and> (\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> \<Inter>\<F> \<noteq> {}) \<longrightarrow> \<Inter>\<U> \<noteq> {})"
   (is "_ = ?rhs")
proof (cases "topspace X = {}")
  case True
  then show ?thesis
    apply (clarsimp simp add: compact_space_def closedin_topspace_empty)
    by (metis finite.emptyI finite_insert infinite_super insertI1 subsetI)
next
  case False
  show ?thesis
  proof safe
    fix \<U> :: "'a set set"
    assume * [rule_format]: "\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> \<Inter>\<F> \<noteq> {}"
    define \<V> where "\<V> \<equiv> (\<lambda>S. topspace X - S) ` \<U>"
    assume clo: "\<forall>C\<in>\<U>. closedin X C" and [simp]: "\<Inter>\<U> = {}"
    then have "\<forall>V \<in> \<V>. openin X V" "topspace X \<subseteq> \<Union>\<V>"
      by (auto simp: \<V>_def)
    moreover assume [unfolded compact_space_alt, rule_format, of \<V>]: "compact_space X"
    ultimately obtain \<F> where \<F>: "finite \<F>" "\<F> \<subseteq> \<U>" "topspace X \<subseteq> topspace X - \<Inter>\<F>"
      by (auto simp: exists_finite_subset_image \<V>_def)
    moreover have "\<F> \<noteq> {}"
      using \<F> \<open>topspace X \<noteq> {}\<close> by blast
    ultimately show "False"
      using * [of \<F>]
      by auto (metis Diff_iff Inter_iff clo closedin_def subsetD)
  next
    assume R [rule_format]: ?rhs
    show "compact_space X"
      unfolding compact_space_alt
    proof clarify
      fix \<U> :: "'a set set"
      define \<V> where "\<V> \<equiv> (\<lambda>S. topspace X - S) ` \<U>"
      assume "\<forall>C\<in>\<U>. openin X C" and "topspace X \<subseteq> \<Union>\<U>"
      with \<open>topspace X \<noteq> {}\<close> have *: "\<forall>V \<in> \<V>. closedin X V" "\<U> \<noteq> {}"
        by (auto simp: \<V>_def)
      show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union>\<F>"
      proof (rule ccontr; simp)
        assume "\<forall>\<F>\<subseteq>\<U>. finite \<F> \<longrightarrow> \<not> topspace X \<subseteq> \<Union>\<F>"
        then have "\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<V> \<longrightarrow> \<Inter>\<F> \<noteq> {}"
          by (simp add: \<V>_def all_finite_subset_image)
        with \<open>topspace X \<subseteq> \<Union>\<U>\<close> show False
          using R [of \<V>] * by (simp add: \<V>_def)
      qed
    qed
  qed
qed

corollary compactin_fip:
  "compactin X S \<longleftrightarrow>
    S \<subseteq> topspace X \<and>
    (\<forall>\<U>. (\<forall>C\<in>\<U>. closedin X C) \<and> (\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> S \<inter> \<Inter>\<F> \<noteq> {}) \<longrightarrow> S \<inter> \<Inter>\<U> \<noteq> {})"
proof (cases "S = {}")
  case False
  show ?thesis
  proof (cases "S \<subseteq> topspace X")
    case True
    then have "compactin X S \<longleftrightarrow>
          (\<forall>\<U>. \<U> \<subseteq> (\<lambda>T. S \<inter> T) ` {T. closedin X T} \<longrightarrow>
           (\<forall>\<F>. finite \<F> \<longrightarrow> \<F> \<subseteq> \<U> \<longrightarrow> \<Inter>\<F> \<noteq> {}) \<longrightarrow> \<Inter>\<U> \<noteq> {})"
      by (simp add: compact_space_fip compactin_subspace closedin_subtopology image_def subset_eq Int_commute imp_conjL)
    also have "\<dots> = (\<forall>\<U>\<subseteq>Collect (closedin X). (\<forall>\<F>. finite \<F> \<longrightarrow> \<F> \<subseteq> (\<inter>) S ` \<U> \<longrightarrow> \<Inter>\<F> \<noteq> {}) \<longrightarrow> \<Inter> ((\<inter>) S ` \<U>) \<noteq> {})"
      by (simp add: all_subset_image)
    also have "\<dots> = (\<forall>\<U>. (\<forall>C\<in>\<U>. closedin X C) \<and> (\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> S \<inter> \<Inter>\<F> \<noteq> {}) \<longrightarrow> S \<inter> \<Inter>\<U> \<noteq> {})"
    proof -
      have eq: "((\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> \<Inter> ((\<inter>) S ` \<F>) \<noteq> {}) \<longrightarrow> \<Inter> ((\<inter>) S ` \<U>) \<noteq> {}) \<longleftrightarrow>
                ((\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> S \<inter> \<Inter>\<F> \<noteq> {}) \<longrightarrow> S \<inter> \<Inter>\<U> \<noteq> {})"  for \<U>
        by simp (use \<open>S \<noteq> {}\<close> in blast)
      show ?thesis
        apply (simp only: imp_conjL [symmetric] all_finite_subset_image eq)
        apply (simp add: subset_eq)
        done
    qed
    finally show ?thesis
      using True by simp
  qed (simp add: compactin_subspace)
qed force

corollary compact_space_imp_nest:
  fixes C :: "nat \<Rightarrow> 'a set"
  assumes "compact_space X" and clo: "\<And>n. closedin X (C n)"
    and ne: "\<And>n. C n \<noteq> {}" and inc: "\<And>m n. m \<le> n \<Longrightarrow> C n \<subseteq> C m"
  shows "(\<Inter>n. C n) \<noteq> {}"
proof -
  let ?\<U> = "range (\<lambda>n. \<Inter>m \<le> n. C m)"
  have "closedin X A" if "A \<in> ?\<U>" for A
    using that clo by auto
  moreover have "(\<Inter>n\<in>K. \<Inter>m \<le> n. C m) \<noteq> {}" if "finite K" for K
  proof -
    obtain n where "\<And>k. k \<in> K \<Longrightarrow> k \<le> n"
      using Max.coboundedI \<open>finite K\<close> by blast
    with inc have "C n \<subseteq> (\<Inter>n\<in>K. \<Inter>m \<le> n. C m)"
    by blast
  with ne [of n] show ?thesis
    by blast
  qed
  ultimately show ?thesis
    using \<open>compact_space X\<close> [unfolded compact_space_fip, rule_format, of ?\<U>]
    by (simp add: all_finite_subset_image INT_extend_simps UN_atMost_UNIV del: INT_simps)
qed

lemma compactin_discrete_topology:
   "compactin (discrete_topology X) S \<longleftrightarrow> S \<subseteq> X \<and> finite S" (is "?lhs = ?rhs")
proof (intro iffI conjI)
  assume L: ?lhs
  then show "S \<subseteq> X"
    by (auto simp: compactin_def)
  have *: "\<And>\<U>. Ball \<U> (openin (discrete_topology X)) \<and> S \<subseteq> \<Union>\<U> \<Longrightarrow>
        (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union>\<F>)"
    using L by (auto simp: compactin_def)
  show "finite S"
    using * [of "(\<lambda>x. {x}) ` X"] \<open>S \<subseteq> X\<close>
    by clarsimp (metis UN_singleton finite_subset_image infinite_super)
next
  assume ?rhs
  then show ?lhs
    by (simp add: finite_imp_compactin)
qed

lemma compact_space_discrete_topology: "compact_space(discrete_topology X) \<longleftrightarrow> finite X"
  by (simp add: compactin_discrete_topology compact_space_def)

lemma compact_space_imp_Bolzano_Weierstrass:
  assumes "compact_space X" "infinite S" "S \<subseteq> topspace X"
  shows "X derived_set_of S \<noteq> {}"
proof
  assume X: "X derived_set_of S = {}"
  then have "closedin X S"
    by (simp add: closedin_contains_derived_set assms)
  then have "compactin X S"
    by (rule closedin_compact_space [OF \<open>compact_space X\<close>])
  with X show False
    by (metis \<open>infinite S\<close> compactin_subspace compact_space_discrete_topology inf_bot_right subtopology_eq_discrete_topology_eq)
qed

lemma compactin_imp_Bolzano_Weierstrass:
   "\<lbrakk>compactin X S; infinite T \<and> T \<subseteq> S\<rbrakk> \<Longrightarrow> S \<inter> X derived_set_of T \<noteq> {}"
  using compact_space_imp_Bolzano_Weierstrass [of "subtopology X S"]
  by (simp add: compactin_subspace derived_set_of_subtopology inf_absorb2 topspace_subtopology)

lemma compact_closure_of_imp_Bolzano_Weierstrass:
   "\<lbrakk>compactin X (X closure_of S); infinite T; T \<subseteq> S; T \<subseteq> topspace X\<rbrakk> \<Longrightarrow> X derived_set_of T \<noteq> {}"
  using closure_of_mono closure_of_subset compactin_imp_Bolzano_Weierstrass by fastforce

lemma discrete_compactin_eq_finite:
   "S \<inter> X derived_set_of S = {} \<Longrightarrow> compactin X S \<longleftrightarrow> S \<subseteq> topspace X \<and> finite S"
  apply (rule iffI)
  using compactin_imp_Bolzano_Weierstrass compactin_subset_topspace apply blast
  by (simp add: finite_imp_compactin_eq)

lemma discrete_compact_space_eq_finite:
   "X derived_set_of (topspace X) = {} \<Longrightarrow> (compact_space X \<longleftrightarrow> finite(topspace X))"
  by (metis compact_space_discrete_topology discrete_topology_unique_derived_set)

lemma image_compactin:
  assumes cpt: "compactin X S" and cont: "continuous_map X Y f"
  shows "compactin Y (f ` S)"
  unfolding compactin_def
proof (intro conjI allI impI)
  show "f ` S \<subseteq> topspace Y"
    using compactin_subset_topspace cont continuous_map_image_subset_topspace cpt by blast
next
  fix \<U> :: "'b set set"
  assume \<U>: "Ball \<U> (openin Y) \<and> f ` S \<subseteq> \<Union>\<U>"
  define \<V> where "\<V> \<equiv> (\<lambda>U. {x \<in> topspace X. f x \<in> U}) ` \<U>"
  have "S \<subseteq> topspace X"
    and *: "\<And>\<U>. \<lbrakk>\<forall>U\<in>\<U>. openin X U; S \<subseteq> \<Union>\<U>\<rbrakk> \<Longrightarrow> \<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> S \<subseteq> \<Union>\<F>"
    using cpt by (auto simp: compactin_def)
  obtain \<F> where \<F>: "finite \<F>" "\<F> \<subseteq> \<V>" "S \<subseteq> \<Union>\<F>"
  proof -
    have 1: "\<forall>U\<in>\<V>. openin X U"
      unfolding \<V>_def using \<U> cont[unfolded continuous_map] by blast
    have 2: "S \<subseteq> \<Union>\<V>"
      unfolding \<V>_def using compactin_subset_topspace cpt \<U> by fastforce
    show thesis
      using * [OF 1 2] that by metis
  qed
  have "\<forall>v \<in> \<V>. \<exists>U. U \<in> \<U> \<and> v = {x \<in> topspace X. f x \<in> U}"
    using \<V>_def by blast
  then obtain U where U: "\<forall>v \<in> \<V>. U v \<in> \<U> \<and> v = {x \<in> topspace X. f x \<in> U v}"
    by metis
  show "\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> f ` S \<subseteq> \<Union>\<F>"
  proof (intro conjI exI)
    show "finite (U ` \<F>)"
      by (simp add: \<open>finite \<F>\<close>)
  next
    show "U ` \<F> \<subseteq> \<U>"
      using \<open>\<F> \<subseteq> \<V>\<close> U by auto
  next
    show "f ` S \<subseteq> \<Union> (U ` \<F>)"
      using \<F>(2-3) U UnionE subset_eq U by fastforce
  qed
qed


lemma homeomorphic_compact_space:
  assumes "X homeomorphic_space Y"
  shows "compact_space X \<longleftrightarrow> compact_space Y"
    using homeomorphic_space_sym
    by (metis assms compact_space_def homeomorphic_eq_everything_map homeomorphic_space image_compactin)

lemma homeomorphic_map_compactness:
  assumes hom: "homeomorphic_map X Y f" and U: "U \<subseteq> topspace X"
  shows "compactin Y (f ` U) \<longleftrightarrow> compactin X U"
proof -
  have "f ` U \<subseteq> topspace Y"
    using hom U homeomorphic_imp_surjective_map by blast
  moreover have "homeomorphic_map (subtopology X U) (subtopology Y (f ` U)) f"
    using U hom homeomorphic_imp_surjective_map by (blast intro: homeomorphic_map_subtopologies)
  then have "compact_space (subtopology Y (f ` U)) = compact_space (subtopology X U)"
    using homeomorphic_compact_space homeomorphic_map_imp_homeomorphic_space by blast
  ultimately show ?thesis
    by (simp add: compactin_subspace U)
qed

lemma homeomorphic_map_compactness_eq:
   "homeomorphic_map X Y f
        \<Longrightarrow> compactin X U \<longleftrightarrow> U \<subseteq> topspace X \<and> compactin Y (f ` U)"
  by (meson compactin_subset_topspace homeomorphic_map_compactness)


subsection\<open>Embedding maps\<close>

definition embedding_map
  where "embedding_map X Y f \<equiv> homeomorphic_map X (subtopology Y (f ` (topspace X))) f"

lemma embedding_map_eq:
   "\<lbrakk>embedding_map X Y f; \<And>x. x \<in> topspace X \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> embedding_map X Y g"
  unfolding embedding_map_def
  by (metis homeomorphic_map_eq image_cong)

lemma embedding_map_compose:
  assumes "embedding_map X X' f" "embedding_map X' X'' g"
  shows "embedding_map X X'' (g \<circ> f)"
proof -
  have hm: "homeomorphic_map X (subtopology X' (f ` topspace X)) f" "homeomorphic_map X' (subtopology X'' (g ` topspace X')) g"
    using assms by (auto simp: embedding_map_def)
  then obtain C where "g ` topspace X' \<inter> C = (g \<circ> f) ` topspace X"
    by (metis (no_types) Int_absorb1 continuous_map_image_subset_topspace continuous_map_in_subtopology homeomorphic_eq_everything_map image_comp image_mono)
  then have "homeomorphic_map (subtopology X' (f ` topspace X)) (subtopology X'' ((g \<circ> f) ` topspace X)) g"
    by (metis hm homeomorphic_imp_surjective_map homeomorphic_map_subtopologies image_comp subtopology_subtopology topspace_subtopology)
  then show ?thesis
  unfolding embedding_map_def
  using hm(1) homeomorphic_map_compose by blast
qed

lemma surjective_embedding_map:
   "embedding_map X Y f \<and> f ` (topspace X) = topspace Y \<longleftrightarrow> homeomorphic_map X Y f"
  by (force simp: embedding_map_def homeomorphic_eq_everything_map)

lemma embedding_map_in_subtopology:
   "embedding_map X (subtopology Y S) f \<longleftrightarrow> embedding_map X Y f \<and> f ` (topspace X) \<subseteq> S"
  apply (auto simp: embedding_map_def subtopology_subtopology Int_absorb1)
    apply (metis (no_types) homeomorphic_imp_surjective_map subtopology_subtopology subtopology_topspace topspace_subtopology)
  apply (simp add: continuous_map_def homeomorphic_eq_everything_map topspace_subtopology)
  done

lemma injective_open_imp_embedding_map:
   "\<lbrakk>continuous_map X Y f; open_map X Y f; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> embedding_map X Y f"
  unfolding embedding_map_def
  apply (rule bijective_open_imp_homeomorphic_map)
  using continuous_map_in_subtopology apply blast
    apply (auto simp: continuous_map_in_subtopology open_map_into_subtopology topspace_subtopology continuous_map)
  done

lemma injective_closed_imp_embedding_map:
  "\<lbrakk>continuous_map X Y f; closed_map X Y f; inj_on f (topspace X)\<rbrakk> \<Longrightarrow> embedding_map X Y f"
  unfolding embedding_map_def
  apply (rule bijective_closed_imp_homeomorphic_map)
     apply (simp_all add: continuous_map_into_subtopology closed_map_into_subtopology)
  apply (simp add: continuous_map inf.absorb_iff2 topspace_subtopology)
  done

lemma embedding_map_imp_homeomorphic_space:
   "embedding_map X Y f \<Longrightarrow> X homeomorphic_space (subtopology Y (f ` (topspace X)))"
  unfolding embedding_map_def
  using homeomorphic_space by blast

subsection \<open>Continuity\<close>

lemma continuous_on_open:
  "continuous_on S f \<longleftrightarrow>
    (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
      openin (subtopology euclidean S) (S \<inter> f -` T))"
  unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
  by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

lemma continuous_on_closed:
  "continuous_on S f \<longleftrightarrow>
    (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
      closedin (subtopology euclidean S) (S \<inter> f -` T))"
  unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
  by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

lemma continuous_on_imp_closedin:
  assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
  shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
  using assms continuous_on_closed by blast

subsection%unimportant \<open>Half-global and completely global cases\<close>

lemma continuous_openin_preimage_gen:
  assumes "continuous_on S f"  "open T"
  shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
proof -
  have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
    by auto
  have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
    using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
  then show ?thesis
    using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
    using * by auto
qed

lemma continuous_closedin_preimage:
  assumes "continuous_on S f" and "closed T"
  shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
proof -
  have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
    by auto
  have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
    using closedin_closed_Int[of T "f ` S", OF assms(2)]
    by (simp add: Int_commute)
  then show ?thesis
    using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T \<inter> f ` S"]]
    using * by auto
qed

lemma continuous_openin_preimage_eq:
   "continuous_on S f \<longleftrightarrow>
    (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
apply safe
apply (simp add: continuous_openin_preimage_gen)
apply (fastforce simp add: continuous_on_open openin_open)
done

lemma continuous_closedin_preimage_eq:
   "continuous_on S f \<longleftrightarrow>
    (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
apply safe
apply (simp add: continuous_closedin_preimage)
apply (fastforce simp add: continuous_on_closed closedin_closed)
done

lemma continuous_open_preimage:
  assumes contf: "continuous_on S f" and "open S" "open T"
  shows "open (S \<inter> f -` T)"
proof-
  obtain U where "open U" "(S \<inter> f -` T) = S \<inter> U"
    using continuous_openin_preimage_gen[OF contf \<open>open T\<close>]
    unfolding openin_open by auto
  then show ?thesis
    using open_Int[of S U, OF \<open>open S\<close>] by auto
qed

lemma continuous_closed_preimage:
  assumes contf: "continuous_on S f" and "closed S" "closed T"
  shows "closed (S \<inter> f -` T)"
proof-
  obtain U where "closed U" "(S \<inter> f -` T) = S \<inter> U"
    using continuous_closedin_preimage[OF contf \<open>closed T\<close>]
    unfolding closedin_closed by auto
  then show ?thesis using closed_Int[of S U, OF \<open>closed S\<close>] by auto
qed

lemma continuous_open_vimage: "open S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> open (f -` S)"
  by (metis continuous_on_eq_continuous_within open_vimage) 
 
lemma continuous_closed_vimage: "closed S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> closed (f -` S)"
  by (simp add: closed_vimage continuous_on_eq_continuous_within)

lemma Times_in_interior_subtopology:
  assumes "(x, y) \<in> U" "openin (subtopology euclidean (S \<times> T)) U"
  obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
                    "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
proof -
  from assms obtain E where "open E" "U = S \<times> T \<inter> E" "(x, y) \<in> E" "x \<in> S" "y \<in> T"
    by (auto simp: openin_open)
  from open_prod_elim[OF \<open>open E\<close> \<open>(x, y) \<in> E\<close>]
  obtain E1 E2 where "open E1" "open E2" "(x, y) \<in> E1 \<times> E2" "E1 \<times> E2 \<subseteq> E"
    by blast
  show ?thesis
  proof
    show "openin (subtopology euclidean S) (E1 \<inter> S)"
      "openin (subtopology euclidean T) (E2 \<inter> T)"
      using \<open>open E1\<close> \<open>open E2\<close>
      by (auto simp: openin_open)
    show "x \<in> E1 \<inter> S" "y \<in> E2 \<inter> T"
      using \<open>(x, y) \<in> E1 \<times> E2\<close> \<open>x \<in> S\<close> \<open>y \<in> T\<close> by auto
    show "(E1 \<inter> S) \<times> (E2 \<inter> T) \<subseteq> U"
      using \<open>E1 \<times> E2 \<subseteq> E\<close> \<open>U = _\<close>
      by (auto simp: )
  qed
qed

lemma closedin_Times:
  "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
    closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
  unfolding closedin_closed using closed_Times by blast

lemma openin_Times:
  "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
    openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
  unfolding openin_open using open_Times by blast

lemma openin_Times_eq:
  fixes S :: "'a::topological_space set" and T :: "'b::topological_space set"
  shows
    "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
      S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
    (is "?lhs = ?rhs")
proof (cases "S' = {} \<or> T' = {}")
  case True
  then show ?thesis by auto
next
  case False
  then obtain x y where "x \<in> S'" "y \<in> T'"
    by blast
  show ?thesis
  proof
    assume ?lhs
    have "openin (subtopology euclidean S) S'"
      apply (subst openin_subopen, clarify)
      apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
      using \<open>y \<in> T'\<close>
       apply auto
      done
    moreover have "openin (subtopology euclidean T) T'"
      apply (subst openin_subopen, clarify)
      apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
      using \<open>x \<in> S'\<close>
       apply auto
      done
    ultimately show ?rhs
      by simp
  next
    assume ?rhs
    with False show ?lhs
      by (simp add: openin_Times)
  qed
qed

lemma Lim_transform_within_openin:
  assumes f: "(f \<longlongrightarrow> l) (at a within T)"
    and "openin (subtopology euclidean T) S" "a \<in> S"
    and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
  shows "(g \<longlongrightarrow> l) (at a within T)"
proof -
  have "\<forall>\<^sub>F x in at a within T. x \<in> T \<and> x \<noteq> a"
    by (simp add: eventually_at_filter)
  moreover
  from \<open>openin _ _\<close> obtain U where "open U" "S = T \<inter> U"
    by (auto simp: openin_open)
  then have "a \<in> U" using \<open>a \<in> S\<close> by auto
  from topological_tendstoD[OF tendsto_ident_at \<open>open U\<close> \<open>a \<in> U\<close>]
  have "\<forall>\<^sub>F x in at a within T. x \<in> U" by auto
  ultimately
  have "\<forall>\<^sub>F x in at a within T. f x = g x"
    by eventually_elim (auto simp: \<open>S = _\<close> eq)
  then show ?thesis using f
    by (rule Lim_transform_eventually)
qed

lemma continuous_on_open_gen:
  assumes "f ` S \<subseteq> T"
    shows "continuous_on S f \<longleftrightarrow>
             (\<forall>U. openin (subtopology euclidean T) U
                  \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    by (clarsimp simp add: continuous_openin_preimage_eq openin_open)
      (metis Int_assoc assms image_subset_iff_subset_vimage inf.absorb_iff1)
next
  assume R [rule_format]: ?rhs
  show ?lhs
  proof (clarsimp simp add: continuous_openin_preimage_eq)
    fix U::"'a set"
    assume "open U"
    then have "openin (subtopology euclidean S) (S \<inter> f -` (U \<inter> T))"
      by (metis R inf_commute openin_open)
    then show "openin (subtopology euclidean S) (S \<inter> f -` U)"
      by (metis Int_assoc Int_commute assms image_subset_iff_subset_vimage inf.absorb_iff2 vimage_Int)
  qed
qed

lemma continuous_openin_preimage:
  "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
        \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
  by (simp add: continuous_on_open_gen)

lemma continuous_on_closed_gen:
  assumes "f ` S \<subseteq> T"
  shows "continuous_on S f \<longleftrightarrow>
             (\<forall>U. closedin (subtopology euclidean T) U
                  \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
    (is "?lhs = ?rhs")
proof -
  have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
    using assms by blast
  show ?thesis
  proof
    assume L: ?lhs
    show ?rhs
    proof clarify
      fix U
      assume "closedin (subtopology euclidean T) U"
      then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
        using L unfolding continuous_on_open_gen [OF assms]
        by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
    qed
  next
    assume R [rule_format]: ?rhs
    show ?lhs
      unfolding continuous_on_open_gen [OF assms]
      by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
  qed
qed

lemma continuous_closedin_preimage_gen:
  assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
    shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
using assms continuous_on_closed_gen by blast

lemma continuous_transform_within_openin:
  assumes "continuous (at a within T) f"
    and "openin (subtopology euclidean T) S" "a \<in> S"
    and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
  shows "continuous (at a within T) g"
  using assms by (simp add: Lim_transform_within_openin continuous_within)


end