--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Jan 14 17:29:04 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Mon Jan 14 17:30:36 2013 +0100
@@ -93,6 +93,25 @@
end
+lemma topological_basis_prod:
+ assumes A: "topological_basis A" and B: "topological_basis B"
+ shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
+ unfolding topological_basis_def
+proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
+ fix S :: "('a \<times> 'b) set" assume "open S"
+ then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
+ proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
+ fix x y assume "(x, y) \<in> S"
+ from open_prod_elim[OF `open S` this]
+ obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
+ by (metis mem_Sigma_iff)
+ moreover from topological_basisE[OF A a] guess A0 .
+ moreover from topological_basisE[OF B b] guess B0 .
+ ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
+ by (intro UN_I[of "(A0, B0)"]) auto
+ qed auto
+qed (metis A B topological_basis_open open_Times)
+
subsection {* Countable Basis *}
locale countable_basis =
@@ -227,6 +246,17 @@
sublocale second_countable_topology < countable_basis "SOME B. countable B \<and> topological_basis B"
using someI_ex[OF ex_countable_basis] by unfold_locales safe
+instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
+proof
+ obtain A :: "'a set set" where "countable A" "topological_basis A"
+ using ex_countable_basis by auto
+ moreover
+ obtain B :: "'b set set" where "countable B" "topological_basis B"
+ using ex_countable_basis by auto
+ ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> topological_basis B"
+ by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod)
+qed
+
subsection {* Polish spaces *}
text {* Textbooks define Polish spaces as completely metrizable.
--- a/src/HOL/Probability/Borel_Space.thy Mon Jan 14 17:29:04 2013 +0100
+++ b/src/HOL/Probability/Borel_Space.thy Mon Jan 14 17:30:36 2013 +0100
@@ -153,36 +153,6 @@
"borel = sigma UNIV union_closed_basis"
by (rule borel_eq_countable_basis[OF countable_union_closed_basis basis_union_closed_basis])
-lemma topological_basis_prod:
- assumes A: "topological_basis A" and B: "topological_basis B"
- shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
- unfolding topological_basis_def
-proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
- fix S :: "('a \<times> 'b) set" assume "open S"
- then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
- proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
- fix x y assume "(x, y) \<in> S"
- from open_prod_elim[OF `open S` this]
- obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
- by (metis mem_Sigma_iff)
- moreover from topological_basisE[OF A a] guess A0 .
- moreover from topological_basisE[OF B b] guess B0 .
- ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
- by (intro UN_I[of "(A0, B0)"]) auto
- qed auto
-qed (metis A B topological_basis_open open_Times)
-
-instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
-proof
- obtain A :: "'a set set" where "countable A" "topological_basis A"
- using ex_countable_basis by auto
- moreover
- obtain B :: "'b set set" where "countable B" "topological_basis B"
- using ex_countable_basis by auto
- ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> topological_basis B"
- by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod)
-qed
-
lemma borel_measurable_Pair[measurable (raw)]:
fixes f :: "'a \<Rightarrow> 'b::second_countable_topology" and g :: "'a \<Rightarrow> 'c::second_countable_topology"
assumes f[measurable]: "f \<in> borel_measurable M"