Let us say for an arbitrary topological space $X$ that it has property $\dagger$ if for any closed equivalence relation $\sim$ on $X$ (closed as a subset of $X^2$), the quotient space $X/{\sim}$ is Hausdorff.
Is there a more "intrinsic" characterisation of property $\dagger$ (in terms of separation axioms of some sort, perhaps)?
I believe compact (possibly non-Hausdorff) spaces have $\dagger$, as in their case for a closed $\sim$, the quotient map is closed, and consequently the product $X^2\to (X/{\sim})^2$ is also closed (and sends $\sim$ to the diagonal).
On the other hand, it is easy to see that if $X$ is Hausdorff and has this property, then it must be normal: otherwise the equivalence relation that identifies all points in two supposedly inseparable closed sets (separately) and leaves all others untouched will be closed will have a non-Hausdorff quotient.
(As a side note, this is true for arbitrary (not neccessarily Hausdorff) topological groups if we restrict $\sim$ to be the relation of lying in the same coset of a subgroup, because the quotient map is open in this case.)
Edit: I've been looking for some references about group actions and stumbled upon a "Lemma" in Duistermat and Kolk's "Lie groups" book which states that $M/{\sim}$ is Hausdorff iff $\sim$ is closed. This is not true (as shown by the above example in a non-$T_4$ space) and I believe I see the mistake made by the authors (they seem to have assumed that the quotient mapping is open), but it has reminded me of this question (about which I'm still curious).