Given a topological space [itex]X[/itex] and a subspace [itex]Y \subseteq X[/itex] let [itex]X / Y[/itex] be the union of the complement [itex]X \backslash Y[/itex] and a set with one point. Define an equivalence relation [itex]\tilde[/itex] on [itex]X[/itex] such that X/~=X/Y, and use it to deifne a topology on [itex]X / Y[/itex] as an identification space of [itex]X[/itex], with projection [itex]p: X \rightarrow X / Y[/itex]. Prove that a subset [itex]Z \subseteq X[/itex] is such that [itex]p(Z)[/itex] is open in [itex]X / Y[/itex] if and only if [itex]Y \cup Z[/itex] is open in [itex]X[/itex]. ok. so i thought the equivalence relation should identify points outside Y wiht points inside Y as that way the quotient would be equal to the complement. but i want the quotient to be equal to X/Y and because its the union of the copmlement and this singleton, i'm confused - this extra one point set is making it hard to see a relation.