I was wondering if there are some necessary and sufficient conditions for the quotient space to be Haussdorf. I have been trying a little for a while, but I only got very restrictive sufficient conditions.



If X is a Haussdorf topological space and R and equivalence relation on X, when is X/R Haussdorf?
If $X$ is Hausdorff *and* the quotient map $X\to X/R$ is open, then $X/R$ is Hausdorff if and only if $R\subseteq X\times X$ is closed (see <https://math.stackexchange.com/questions/91639/x-sim-is-hausdorff-if-and-only-if-sim-is-closed-in-x-times-x>).


