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products/sources/formale Sprachen/Isabelle/HOL/Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 7 kB

Quelle Sum_Topology.thy Sprache: Isabelle

section\<open>Disjoint sum of arbitarily many spaces\<close>

theory Sum_Topology
  imports Abstract_Topology
begin

definition sum_topology :: "('a \ 'b topology) \ 'a set \ ('a \ 'b) topology" where
  "sum_topology X I \
    topology (\<lambda>U. U \<subseteq> Sigma I (topspace \<circ> X) \<and> (\<forall>i \<in> I. openin (X i) {x. (i,x) \<in> U}))"

lemma is_sum_topology: "istopology (\U. U \ Sigma I (topspace \ X) \ (\i\I. openin (X i) {x. (i, x) \ U}))"
proof -
  have 1: "{x. (i, x) \ S \ T} = {x. (i, x) \ S} \ {x::'b. (i, x) \ T}" for S T and i::'a
    by auto
  have 2: "{x. (i, x) \ \ \} = (\K\\. {x::'b. (i, x) \ K})" for \ and i::'a
    by auto
  show ?thesis
    unfolding istopology_def 1 2 by blast
qed

lemma openin_sum_topology:
   "openin (sum_topology X I) U \
        U \<subseteq> Sigma I (topspace \<circ> X) \<and> (\<forall>i \<in> I. openin (X i) {x. (i,x) \<in> U})"
  by (auto simp: sum_topology_def is_sum_topology)

lemma openin_disjoint_union:
   "openin (sum_topology X I) (Sigma I U) \ (\i \ I. openin (X i) (U i))"
  using openin_subset by (force simp: openin_sum_topology)

lemma topspace_sum_topology [simp]:
   "topspace(sum_topology X I) = Sigma I (topspace \ X)"
  by (metis comp_apply openin_disjoint_union openin_subset openin_sum_topology openin_topspace subset_antisym)

lemma openin_sum_topology_alt:
   "openin (sum_topology X I) U \ (\T. U = Sigma I T \ (\i \ I. openin (X i) (T i)))"
  by (bestsimp simp: openin_sum_topology dest: openin_subset)

lemma forall_openin_sum_topology:
   "(\U. openin (sum_topology X I) U \ P U) \ (\T. (\i \ I. openin (X i) (T i)) \ P(Sigma I T))"
  by (auto simp: openin_sum_topology_alt)

lemma exists_openin_sum_topology:
   "(\U. openin (sum_topology X I) U \ P U) \
    (\<exists>T. (\<forall>i \<in> I. openin (X i) (T i)) \<and> P(Sigma I T))"
  by (auto simp: openin_sum_topology_alt)

lemma closedin_sum_topology:
   "closedin (sum_topology X I) U \ U \ Sigma I (topspace \ X) \ (\i \ I. closedin (X i) {x. (i,x) \ U})"
     (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  then have U: "U \ Sigma I (topspace \ X)"
    using closedin_subset by fastforce
  then have "\i\I. {x. (i, x) \ U} \ topspace (X i)"
    by fastforce
  moreover have "openin (X i) (topspace (X i) - {x. (i, x) \ U})" if "i\I" for i
    apply (subst openin_subopen)
    using L that unfolding closedin_def openin_sum_topology
    by (bestsimp simp: o_def subset_iff)
  ultimately show ?rhs
    by (simp add: U closedin_def)
next
  assume R: ?rhs
  then have "openin (X i) {x. (i, x) \ topspace (sum_topology X I) - U}" if "i\I" for i
    apply (subst openin_subopen)
    using that unfolding closedin_def openin_sum_topology
    by (bestsimp simp: o_def subset_iff)
  then show ?lhs
    by (simp add: R closedin_def openin_sum_topology)
qed

lemma closedin_disjoint_union:
   "closedin (sum_topology X I) (Sigma I U) \ (\i \ I. closedin (X i) (U i))"
  using closedin_subset by (force simp: closedin_sum_topology)

lemma closedin_sum_topology_alt:
   "closedin (sum_topology X I) U \ (\T. U = Sigma I T \ (\i \ I. closedin (X i) (T i)))"
  using closedin_subset unfolding closedin_sum_topology by auto blast

lemma forall_closedin_sum_topology:
   "(\U. closedin (sum_topology X I) U \ P U) \
        (\<forall>t. (\<forall>i \<in> I. closedin (X i) (t i)) \<longrightarrow> P(Sigma I t))"
  by (metis closedin_sum_topology_alt)

lemma exists_closedin_sum_topology:
   "(\U. closedin (sum_topology X I) U \ P U) \
    (\<exists>T. (\<forall>i \<in> I. closedin (X i) (T i)) \<and> P(Sigma I T))"
  by (simp add: closedin_sum_topology_alt) blast

lemma open_map_component_injection:
   "i \ I \ open_map (X i) (sum_topology X I) (\x. (i,x))"
  unfolding open_map_def openin_sum_topology
  using openin_subset openin_subopen by (force simp: image_iff)

lemma closed_map_component_injection:
  assumes "i \ I"
  shows "closed_map(X i) (sum_topology X I) (\x. (i,x))"
proof -
  have "closedin (X j) {x. j = i \ x \ U}"
    if "\U S. closedin U S \ S \ topspace U" and "closedin (X i) U" and "j \ I" for U j
    using that by (cases "j=i") auto
  then show ?thesis
    using closedin_subset assms by (force simp: closed_map_def closedin_sum_topology image_iff)
qed

lemma continuous_map_component_injection:
   "i \ I \ continuous_map(X i) (sum_topology X I) (\x. (i,x))"
  by (auto simp: continuous_map_def openin_sum_topology Collect_conj_eq openin_Int)

lemma subtopology_sum_topology:
  "subtopology (sum_topology X I) (Sigma I S) =
        sum_topology (\<lambda>i. subtopology (X i) (S i)) I"
  unfolding topology_eq
proof (intro strip iffI)
  fix T
  assume *: "openin (subtopology (sum_topology X I) (Sigma I S)) T"
  then obtain U where U: "\i\I. openin (X i) (U i) \ T = Sigma I U \ Sigma I S"
    by (auto simp: openin_subtopology openin_sum_topology_alt)
  have "T = (SIGMA i:I. U i \ S i)"
  proof
    show "T \ (SIGMA i:I. U i \ S i)"
      by (metis "*" SigmaE Sigma_Int_distrib2 U openin_imp_subset subset_iff)
    show "(SIGMA i:I. U i \ S i) \ T"
      using U by fastforce
  qed
  then show "openin (sum_topology (\i. subtopology (X i) (S i)) I) T"
    by (simp add: U openin_disjoint_union openin_subtopology_Int)
next
  fix T
  assume *: "openin (sum_topology (\i. subtopology (X i) (S i)) I) T"
  then obtain U where "\i\I. \T. openin (X i) T \ U i = T \ S i" and Teq: "T = Sigma I U"
    by (auto simp: openin_subtopology openin_sum_topology_alt)
  then obtain B where "\i. i\I \ openin (X i) (B i) \ U i = B i \ S i"
    by metis
  then show "openin (subtopology (sum_topology X I) (Sigma I S)) T"
    by (auto simp: openin_subtopology openin_sum_topology_alt Teq)
qed

lemma embedding_map_component_injection:
   "i \ I \ embedding_map (X i) (sum_topology X I) (\x. (i,x))"
  by (metis injective_open_imp_embedding_map continuous_map_component_injection
            open_map_component_injection inj_onI prod.inject)

lemma topological_property_of_sum_component:
  assumes major: "P (sum_topology X I)"
    and minor: "\X S. \P X; closedin X S; openin X S\ \ P(subtopology X S)"
    and PQ:  "\X Y. X homeomorphic_space Y \ (P X \ Q Y)"
  shows "(\i \ I. Q(X i))"
proof -
  have "Q(X i)" if "i \ I" for i
  proof -
    have "closed_map (X i) (sum_topology X I) (Pair i)"
      by (simp add: closed_map_component_injection that)
    moreover have "open_map (X i) (sum_topology X I) (Pair i)"
      by (simp add: open_map_component_injection that)
    ultimately have "P(subtopology (sum_topology X I) (Pair i ` topspace (X i)))"
      by (simp add: closed_map_def major minor open_map_def)
    then show ?thesis
      by (metis PQ embedding_map_component_injection embedding_map_imp_homeomorphic_space homeomorphic_space_sym that)
  qed
  then show ?thesis by metis
qed

end

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