src/HOL/Analysis/Topology_Euclidean_Space.thy
changeset 66304 cde6ceffcbc7
parent 66286 1c977b13414f
child 66408 46cfd348c373
equal deleted inserted replaced
66303:210dae34b8bc 66304:cde6ceffcbc7
  5298 
  5298 
  5299 lemma compact_eq_seq_compact_metric:
  5299 lemma compact_eq_seq_compact_metric:
  5300   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
  5300   "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
  5301   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
  5301   using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
  5302 
  5302 
  5303 lemma compact_def: --\<open>this is the definition of compactness in HOL Light\<close>
  5303 lemma compact_def: \<comment>\<open>this is the definition of compactness in HOL Light\<close>
  5304   "compact (S :: 'a::metric_space set) \<longleftrightarrow>
  5304   "compact (S :: 'a::metric_space set) \<longleftrightarrow>
  5305    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
  5305    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
  5306   unfolding compact_eq_seq_compact_metric seq_compact_def by auto
  5306   unfolding compact_eq_seq_compact_metric seq_compact_def by auto
  5307 
  5307 
  5308 subsubsection \<open>Complete the chain of compactness variants\<close>
  5308 subsubsection \<open>Complete the chain of compactness variants\<close>