Continuous bijection from compact is a homeomorphism criterion

Homeomorphism criterion (compact to metric): Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces . If $X$ is compact and $f:X\to Y$ is a continuous bijection , then $f^{-1}:f(X)\to X$ is continuous. In particular, $f$ is a homeomorphism from $X$ onto $f(X)$.

This result says that on compact domains, a continuous bijection automatically has a continuous inverse (when the target is Hausdorff, which metric spaces are).

Proof sketch (optional): Let $C\subseteq X$ be closed . Since $X$ is compact, $C$ is compact, hence $f(C)$ is compact in $Y$, hence closed in $Y$. Therefore $f$ maps closed sets to closed sets, which implies $f^{-1}$ is continuous.