TITLE: What are the epimorphisms in the category of Hausdorff spaces?
QUESTION [20 upvotes]: 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?

REPLY [5 votes]: This should be a comment rather than an answer, but I don't have enough rep.  
HTop is actually the largest subcategory of Top closed under finite limits (as computed in Top) where all maps with dense image are epi:   
If $X$ is not Hausdorff then the equalizer of the projections $\pi_{1},\pi_{2}:X\times X\rightarrow X$, which is just the diagonal $\delta:X\rightarrow X\times X$, is not closed.  (Recall that a space is Hausdorff iff the diagonal is closed.)  
Let $C$ denote the closure of the diagonal in $X\times X$, let $d$ denote the factorization of $\delta$ through $C$, and let $p_{1}$ and $p_{2}$ denote the restrictions of $\pi_{1}$ and $\pi_{2}$ to $C$ respectively.  Then the image of $X$ is dense in $C$ and $p_{1}\circ d = p_{2} \circ d$, but $p_{1}\neq p_{2}$, so $d$ is not epi. 
This shows the fact Andy mentions in the comments, that equalizers are closed subspaces in HTop, is essential.