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 @@
1.5  end
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 *}
1.28  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
1.33 +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 *}
1.46  text {* Textbooks define Polish spaces as completely metrizable.