In a compact Hausdorff space, compact subsets are exactly the closed subsets.Suggest a slogan
Let \(X\) be a compact Hausdorff space. A subset \(U\subseteq X\) is closed if and only if it is compact [@Mun00].
Closed subsets of compact spaces are compact [@Mun00], and compact subsets of Hausdorff spaces are closed [@Mun00]. Only the second needs \(X\) Hausdorff, and only the first needs \(X\) compact: \(\RR\) is closed in \(\RR\) and not compact.