It is known that the $T\_0$ and $T\_2$ axioms are not preserved under open, closed and continuous maps (for instance, see here: [An example of open closed continuous image of $T\_0$-space that is not $T\_0$](https://math.stackexchange.com/questions/1411751/an-example-of-open-closed-continuous-image-of-t-0-space-that-is-not-t-0) and here: [An example of open closed continuous image of $T\_2$-space that is not $T\_2$](https://math.stackexchange.com/questions/1412492/an-example-of-open-closed-continuous-image-of-t-2-space-that-is-not-t-2?rq=1%5D)). However, it is not difficult to verify that the Hausdorff property is preserved under perfect functions (closed with compact fibers).


Is it also true that a perfect image of a $T\_0$ space is $T\_0$ as well? It is not difficult to see that a perfect image of a finite $T\_0$ space is indeed $T\_0$. What about infinite $T\_0$ spaces?



Preservation of separation axioms under perfect functions
Yes, it seems to be true.


First, towards a counterexample. Let $f\colon X \to Y$ be a perfect map from a $T\_0$ space. If $Y$ is not $T\_0$, it contains a two-point indiscrete space $B$, and $f\colon f^{-1}[B] \to B$ is also a perfect map. So if there is a counterexaple at all, there is a counterexample realized by a $T\_0$ space $X$ consisting of two compact parts such that $(\*)$: every non-empty closed subset intersects both of them. Necessarily $X$ is compact. (In fact, given $(\*)$, $X$ is compact if and only if both parts are compact since every open cover of one part is an open cover of $X$.)


However, no compact $T\_0$ space satisfying $(\*)$ exists. Every compact space contains a minimal non-empty closed subset by Zorn's lemma. And in a $T\_0$ space, this minimal closed set has to be a singleton.


