proposition: Lemma T0_sep_Subspace {X : TopologicalSpace} (A : Ensemble X) : T0_sep X -> T0_sep (SubspaceTopology A).
proof: Proof. intros HX [x Hx] [y Hy] Hxy. specialize (HX x y) as [[U [HU0 [HU1 HU2]]]|[U [HU0 [HU1 HU2]]]]; [|left|right]. { intros ?. subst. destruct (PI.proof_irrelevance _ Hx Hy). contradiction. } all: exists (inverse_image (subspace_inc A) U); (split; [|split]). 1, 4: apply subspace_inc_continuous; assumption. 1, 4: constructor; simpl; assumption. 1, 2: intros [H0]; simpl in H0; auto. Qed.
imports: Coq(Program.Subset) Topology(Homeomorphisms SubspaceTopology)