It appears to be the case that the epimorphisms in $\text{Haus}$ are precisely the maps with dense image. This is claimed in various places, but a comment on my blog has made me doubt the source I got my proof from (Borceux).
Borceux's argument crucially uses the following result:
If $A \subset X$ is a closed subspace of a Hausdorff space $X$, then the quotient $X/A$ is Hausdorff.
This appears to be false. As far as I can tell, if $X/A$ is Hausdorff, then $A$ and points in $X$ not in $A$ must be separated by open neighborhoods in $X$. But if this is true for every closed subspace $A$ of $X$, then $X$ is necessarily regular, and there are examples of Hausdorff spaces that aren't regular.
So: is it still true that the epimorphisms are precisely the maps with dense image? If so, what is a correct proof of this?