--- a/src/HOL/Analysis/Elementary_Metric_Spaces.thy Mon Dec 02 13:41:44 2019 -0500
+++ b/src/HOL/Analysis/Elementary_Metric_Spaces.thy Mon Dec 02 13:57:32 2019 -0500
@@ -4,7 +4,7 @@
Author: Brian Huffman, Portland State University
*)
-chapter \<open>Functional Analysis\<close>
+section \<open>Elementary Metric Spaces\<close>
theory Elementary_Metric_Spaces
imports
@@ -12,8 +12,6 @@
Metric_Arith
begin
-section \<open>Elementary Metric Spaces\<close>
-
subsection \<open>Open and closed balls\<close>
definition\<^marker>\<open>tag important\<close> ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"