Closed subset of compact set is compact

Closed subset of compact set is compact: Let $X$ be a topological space and let $K\subseteq X$ be compact . If $F\subseteq K$ is closed in the subspace topology on $K$ (equivalently, if $F=K\cap C$ for some closed $C\subseteq X$), then $F$ is compact.

This permanence property is frequently combined with the finite intersection property characterization of compactness.