TOPOLOGY 2026-05-08

Compact Subsets of $\mathbb{R}^n$

Definition

(Compactness)

A subset $K\subseteq\mathbb{R}^n$ is compact if every open cover of $K$ has a finite subcover.

Theorem

(Heine–Borel Theorem)

A subset $K\subseteq\mathbb{R}^n$ is compact if and only if it is both closed and bounded.

This characterisation is special to $\mathbb{R}^n$ . In general metric spaces, compactness is equivalent to sequential compactness (every sequence has a convergent subsequence), but not to being closed and bounded.