Compact-to-Hausdorff homeomorphism criterion

Compact-to-Hausdorff homeomorphism criterion: Let $f:X\to Y$ be a continuous map that is bijective. If $X$ is compact and $Y$ is Hausdorff , then $f$ is a homeomorphism (equivalently, $f^{-1}:Y\to X$ is continuous).

A common way to use this is via the fact that in a Hausdorff space, compact sets are closed ; in particular, under the hypotheses above, images of closed sets in $X$ are closed in $Y$, so $f$ is a closed map.