Quotient topological space ($X/\sim$)

A quotient topological space identifies points of a topological space according to an equivalence relation and equips the set of equivalence classes with the quotient topology. A subset is open exactly when its inverse image under the canonical quotient map is open. Even a Hausdorff space can have a non-Hausdorff quotient; identifying real numbers differing by a rational number gives a multi-point quotient with the indiscrete topology.