Bijective continuous map from compact space to NON HAUSDORFF SPACE prove homeomorphism

Question

Let $X,Y$ be topological spaces with $X$ compact and let $f:X \to Y$ be continuous and bijective. Prove that $f$ is a homeomorphism.

First of all, the image of a compact space under a continuous map is compact. Thus by the bijectivity of $f$, $Y$ is also compact. Now I am not sure, how to proceed. Can anyone give me a hint?

Answer 1

This isn't true. For instance, take any finite set with more than one element and let $X$ be that set with the discrete topology and $Y$ be that set with the indiscrete topology. Then $X$ is compact and the identity map is a continuous bijection $X\to Y$, but it is not a homeomorphism.