Closed equivalence relation with an open quotient map

If an equivalence relation is closed in a space's square and the quotient map is open, the quotient is Hausdorff. An inequivalent pair has an open rectangle avoiding the relation. The two factor images are open quotient neighborhoods and are disjoint, since an intersection would give equivalent representatives in that rectangle. Conversely, a Hausdorff quotient always has a closed equivalence relation by pulling back its closed diagonal.