Continuous Functions from a Compact Space to a Hausdorff Space are Closed Maps

Last updated Nov 1, 2022

Suppose $X$ is a Compact space, $Y$ is a Hausdorff space, and $f: X \to Y$ is continuous. Then $f$ is a Closed Map.

Let $K \subset X$ be Closed. Since a Closed Subset of a Compact Set is Compact, $K$ is Compact. Continuous Functions Preserve Compactness, so $f(K)$ is Compact in $Y$. Compact Sets in Hausdorff Spaces are Closed, so $f(K)$ is Closed. Since $K \subset X$ was an arbitrary Closed set, $f$ is a Closed Map. $\blacksquare$