Consider a Hausdorff topological space $(X,\tau)$. Suppose $(X,\tilde{\tau})$ is the minimal normalization of $(X,{\tau})$, that is, for every given normal topology $\sigma$, where $\tau \subset \sigma$ we have $\tau \subset \tilde{\tau} \subset \sigma$.
Suppose $A\subsetneq X$ and $d$ is a metric function for $(X,\tilde{\tau})$. Let $f(-)=d(A,-):(X,\tau) \to \mathbb{R}$. Induce the final topology $\alpha$ on $\mathbb{R}$ such that $f$ is continuous, is the space $(\mathbb{R},\alpha)$ a Hausdorff space?
While I have stated my main problem, I am unsure about the existence of $\tilde{\tau}$ as the minimal normalization of $\tau$, in case there is a counterexample (which I would be grateful to hear about it), suppose that such $\tilde{\tau}$ exists.
As the problem itself does not seem related to category theory, here is a brief explanation:
If my assumption is correct and the mentioned space is indeed Hausdorff, one can use $f$ and $2f$ to show that for any function $g:Z \to (X,\tau)$ where $g(Z)=A$, when the image of $g$ is not dense in $X$, then $g$ is not epic. That is, in the category of Hausdorff spaces, epis are maps with dense image.