Does a continuous bijection from a compact, hausdorff space imply it is an homeomorphism

Question

Let $f:X \rightarrow Y$ be a bijective, continuous map between two toplogical spaces. Does X being compact and Hausdorff imply that $f$ must be a homeomorphism? I think it doesn't but I can not find an example.

Answer 1

Let $X$ be your favourite compact Hausdorff space with at least $2$ points, let $Y$ be $X$ with the trivial topology, consider the identity as $f$.