Solution
= Solution
A <closed set> $C\subseteq X$ is compact because it is closed in the compact space $X$. Its image $f(C)$ is compact by the fact that a <continuous image of a compact space> is compact, and every compact subset of the <Hausdorff space> $Y$ is closed. Thus the continuous bijection $f$ is a <closed map>. Its inverse $f^{-1}:Y\to X$ is consequently continuous, so $f$ is a <homeomorphism>. This is the <compact-to-Hausdorff continuous bijection theorem>.