Pullback of an epimorphism in the category of Hausdorff spaces

Question

Can anyone give me an example of a pullback of an epimorphism which is not an epimorphism, in the category of Hausdorff spaces?

I've been thinking about it but I have no idea.

Answer 1

Let $X$ be a Hausdorff space and $D$ a proper dense subset of $X$. As you said in the comments, the inclusion map $i \colon D \to X$ is an epimorphism in the category of Hausdorff spaces; but if we take $x \in X \setminus D$ and $f \colon \{*\} \to X$ the map sending $*$ to $x$, then $$ \require{AMScd} \begin{CD} f^{-1}[D] @>f>> D \\ @VjVV @VViV \\[-0.8mm] \{*\} @>>f> X \end{CD} $$ is a pullback square in which $j$ (also the inclusion map) is not an epimorphism (since $f^{-1}[D]=\varnothing$).