src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
changeset 50882 a382bf90867e
parent 50881 ae630bab13da
child 50883 1421884baf5b
     1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Jan 14 17:29:04 2013 +0100
     1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Jan 14 17:30:36 2013 +0100
     1.3 @@ -93,6 +93,25 @@
     1.4  
     1.5  end
     1.6  
     1.7 +lemma topological_basis_prod:
     1.8 +  assumes A: "topological_basis A" and B: "topological_basis B"
     1.9 +  shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
    1.10 +  unfolding topological_basis_def
    1.11 +proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
    1.12 +  fix S :: "('a \<times> 'b) set" assume "open S"
    1.13 +  then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
    1.14 +  proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
    1.15 +    fix x y assume "(x, y) \<in> S"
    1.16 +    from open_prod_elim[OF `open S` this]
    1.17 +    obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
    1.18 +      by (metis mem_Sigma_iff)
    1.19 +    moreover from topological_basisE[OF A a] guess A0 .
    1.20 +    moreover from topological_basisE[OF B b] guess B0 .
    1.21 +    ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
    1.22 +      by (intro UN_I[of "(A0, B0)"]) auto
    1.23 +  qed auto
    1.24 +qed (metis A B topological_basis_open open_Times)
    1.25 +
    1.26  subsection {* Countable Basis *}
    1.27  
    1.28  locale countable_basis =
    1.29 @@ -227,6 +246,17 @@
    1.30  sublocale second_countable_topology < countable_basis "SOME B. countable B \<and> topological_basis B"
    1.31    using someI_ex[OF ex_countable_basis] by unfold_locales safe
    1.32  
    1.33 +instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
    1.34 +proof
    1.35 +  obtain A :: "'a set set" where "countable A" "topological_basis A"
    1.36 +    using ex_countable_basis by auto
    1.37 +  moreover
    1.38 +  obtain B :: "'b set set" where "countable B" "topological_basis B"
    1.39 +    using ex_countable_basis by auto
    1.40 +  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> topological_basis B"
    1.41 +    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod)
    1.42 +qed
    1.43 +
    1.44  subsection {* Polish spaces *}
    1.45  
    1.46  text {* Textbooks define Polish spaces as completely metrizable.