src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 changeset 44365 5daa55003649 parent 44342 8321948340ea child 44457 d366fa5551ef
```     1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sun Aug 21 11:03:15 2011 -0700
1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sun Aug 21 12:22:31 2011 -0700
1.3 @@ -622,6 +622,23 @@
1.4    qed
1.5  qed
1.6
1.7 +lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
1.8 +proof (rule interior_unique)
1.9 +  show "interior A \<times> interior B \<subseteq> A \<times> B"
1.10 +    by (intro Sigma_mono interior_subset)
1.11 +  show "open (interior A \<times> interior B)"
1.12 +    by (intro open_Times open_interior)
1.13 +  show "\<forall>T. T \<subseteq> A \<times> B \<and> open T \<longrightarrow> T \<subseteq> interior A \<times> interior B"
1.14 +    apply (simp add: open_prod_def, clarify)
1.15 +    apply (drule (1) bspec, clarify, rename_tac C D)
1.16 +    apply (simp add: interior_def, rule conjI)
1.17 +    apply (rule_tac x=C in exI, clarsimp)
1.18 +    apply (rule SigmaD1, erule subsetD, erule subsetD, erule (1) SigmaI)
1.19 +    apply (rule_tac x=D in exI, clarsimp)
1.20 +    apply (rule SigmaD2, erule subsetD, erule subsetD, erule (1) SigmaI)
1.21 +    done
1.22 +qed
1.23 +
1.24
1.25  subsection {* Closure of a Set *}
1.26
1.27 @@ -793,6 +810,23 @@
1.28    unfolding closure_interior
1.29    by blast
1.30
1.31 +lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
1.32 +proof (intro closure_unique conjI)
1.33 +  show "A \<times> B \<subseteq> closure A \<times> closure B"
1.34 +    by (intro Sigma_mono closure_subset)
1.35 +  show "closed (closure A \<times> closure B)"
1.36 +    by (intro closed_Times closed_closure)
1.37 +  show "\<forall>T. A \<times> B \<subseteq> T \<and> closed T \<longrightarrow> closure A \<times> closure B \<subseteq> T"
1.38 +    apply (simp add: closed_def open_prod_def, clarify)
1.39 +    apply (rule ccontr)
1.40 +    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
1.41 +    apply (simp add: closure_interior interior_def)
1.42 +    apply (drule_tac x=C in spec)
1.43 +    apply (drule_tac x=D in spec)
1.44 +    apply auto
1.45 +    done
1.46 +qed
1.47 +
1.48
1.49  subsection {* Frontier (aka boundary) *}
1.50
```