src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 changeset 49711 e5aaae7eadc9 parent 48125 602dc0215954 child 49834 b27bbb021df1
1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Oct 05 13:48:22 2012 +0200
1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Oct 05 13:57:48 2012 +0200
1.3 @@ -28,12 +28,14 @@
1.5  lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
1.6  proof-
1.7 -  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
1.8 +  { assume "T1=T2"
1.9 +    hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }
1.10    moreover
1.11 -  {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
1.12 +  { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
1.13      hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
1.14      hence "topology (openin T1) = topology (openin T2)" by simp
1.15 -    hence "T1 = T2" unfolding openin_inverse .}
1.16 +    hence "T1 = T2" unfolding openin_inverse .
1.17 +  }
1.18    ultimately show ?thesis by blast
1.19  qed
1.21 @@ -66,9 +68,11 @@
1.23  lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
1.25 -lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
1.26 +lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
1.27 +  (is "?lhs \<longleftrightarrow> ?rhs")
1.28  proof
1.29 -  assume ?lhs then show ?rhs by auto
1.30 +  assume ?lhs
1.31 +  then show ?rhs by auto
1.32  next
1.33    assume H: ?rhs
1.34    let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
1.35 @@ -77,6 +81,7 @@
1.36    finally show "openin U S" .
1.37  qed
1.39 +
1.40  subsubsection {* Closed sets *}
1.42  definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
1.43 @@ -167,9 +172,11 @@
1.44    apply (rule iffI, clarify)
1.45    apply (frule openin_subset[of U])  apply blast
1.46    apply (rule exI[where x="topspace U"])
1.47 -  by auto
1.48 -
1.49 -lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
1.50 +  apply auto
1.51 +  done
1.52 +
1.53 +lemma subtopology_superset:
1.54 +  assumes UV: "topspace U \<subseteq> V"
1.55    shows "subtopology U V = U"
1.56  proof-
1.57    {fix S