equal
deleted
inserted
replaced
792 |
792 |
793 lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U" |
793 lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U" |
794 by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym) |
794 by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym) |
795 |
795 |
796 lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U" |
796 lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U" |
797 by (simp add: Diff_subset closedin_def) |
797 by (simp add: closedin_def) |
798 |
798 |
799 lemma discrete_topology_unique: |
799 lemma discrete_topology_unique: |
800 "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs") |
800 "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs") |
801 proof |
801 proof |
802 assume R: ?rhs |
802 assume R: ?rhs |