equal
deleted
inserted
replaced
1105 |
1105 |
1106 lemma retract_of_connected: |
1106 lemma retract_of_connected: |
1107 "\<lbrakk>connected T; S retract_of T\<rbrakk> \<Longrightarrow> connected S" |
1107 "\<lbrakk>connected T; S retract_of T\<rbrakk> \<Longrightarrow> connected S" |
1108 by (metis Topological_Spaces.connected_continuous_image retract_of_def retraction) |
1108 by (metis Topological_Spaces.connected_continuous_image retract_of_def retraction) |
1109 |
1109 |
1110 lemma retraction_imp_quotient_map: |
1110 lemma retraction_openin_vimage_iff: |
1111 "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U" |
1111 "openin (top_of_set S) (S \<inter> r -` U) \<longleftrightarrow> openin (top_of_set T) U" |
1112 if retraction: "retraction S T r" and "U \<subseteq> T" |
1112 if retraction: "retraction S T r" and "U \<subseteq> T" |
1113 using retraction apply (rule retractionE) |
1113 using retraction apply (rule retractionE) |
1114 apply (rule continuous_right_inverse_imp_quotient_map [where g=r]) |
1114 apply (rule continuous_right_inverse_imp_quotient_map [where g=r]) |
1115 using \<open>U \<subseteq> T\<close> apply (auto elim: continuous_on_subset) |
1115 using \<open>U \<subseteq> T\<close> apply (auto elim: continuous_on_subset) |