Heine–Borel theorem

Heine–Borel theorem: In $\mathbb{R}^n$ with its usual metric (and induced topology), a set $K\subseteq \mathbb{R}^n$ is compact if and only if it is both closed and bounded .

This specializes general implications like compactness implies boundedness and compactness implies closedness into a complete characterization in Euclidean settings, and it is frequently paired with Bolzano–Weierstrass and sequential criteria such as sequential compactness .