--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 24 15:06:13 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 24 15:32:40 2011 -0700
@@ -7,7 +7,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs" Norm_Arith L2_Norm
begin
(* to be moved elsewhere *)
@@ -20,7 +20,7 @@
apply(subst(2) euclidean_dist_l2) apply(cases "i<DIM('a)")
apply(rule member_le_setL2) by auto
-subsection {* General notion of a topologies as values *}
+subsection {* General notion of a topology as a value *}
definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
typedef (open) 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"