equal
deleted
inserted
replaced
2115 shows "continuous_map X' X'' g" |
2115 shows "continuous_map X' X'' g" |
2116 unfolding quotient_map_def continuous_map_def |
2116 unfolding quotient_map_def continuous_map_def |
2117 proof (intro conjI ballI allI impI) |
2117 proof (intro conjI ballI allI impI) |
2118 show "\<And>x'. x' \<in> topspace X' \<Longrightarrow> g x' \<in> topspace X''" |
2118 show "\<And>x'. x' \<in> topspace X' \<Longrightarrow> g x' \<in> topspace X''" |
2119 using assms unfolding quotient_map_def |
2119 using assms unfolding quotient_map_def |
2120 by (metis (no_types, hide_lams) continuous_map_image_subset_topspace image_comp image_subset_iff) |
2120 by (metis (no_types, opaque_lifting) continuous_map_image_subset_topspace image_comp image_subset_iff) |
2121 next |
2121 next |
2122 fix U'' :: "'c set" |
2122 fix U'' :: "'c set" |
2123 assume U'': "openin X'' U''" |
2123 assume U'': "openin X'' U''" |
2124 have "f ` topspace X = topspace X'" |
2124 have "f ` topspace X = topspace X'" |
2125 by (simp add: f quotient_imp_surjective_map) |
2125 by (simp add: f quotient_imp_surjective_map) |
3800 "retraction_map X Y f \<Longrightarrow> f ` (topspace X) = topspace Y" |
3800 "retraction_map X Y f \<Longrightarrow> f ` (topspace X) = topspace Y" |
3801 by (simp add: retraction_imp_quotient_map quotient_imp_surjective_map) |
3801 by (simp add: retraction_imp_quotient_map quotient_imp_surjective_map) |
3802 |
3802 |
3803 lemma section_imp_injective_map: |
3803 lemma section_imp_injective_map: |
3804 "\<lbrakk>section_map X Y f; x \<in> topspace X; y \<in> topspace X\<rbrakk> \<Longrightarrow> f x = f y \<longleftrightarrow> x = y" |
3804 "\<lbrakk>section_map X Y f; x \<in> topspace X; y \<in> topspace X\<rbrakk> \<Longrightarrow> f x = f y \<longleftrightarrow> x = y" |
3805 by (metis (mono_tags, hide_lams) retraction_maps_def section_map_def) |
3805 by (metis (mono_tags, opaque_lifting) retraction_maps_def section_map_def) |
3806 |
3806 |
3807 lemma retraction_maps_to_retract_maps: |
3807 lemma retraction_maps_to_retract_maps: |
3808 "retraction_maps X Y r s |
3808 "retraction_maps X Y r s |
3809 \<Longrightarrow> retraction_maps X (subtopology X (s ` (topspace Y))) (s \<circ> r) id" |
3809 \<Longrightarrow> retraction_maps X (subtopology X (s ` (topspace Y))) (s \<circ> r) id" |
3810 unfolding retraction_maps_def |
3810 unfolding retraction_maps_def |