Compactness Basics

Compactness is a topological property ensuring a space can be "covered" by a finite number of open sets, generalizing boundedness.

Definition

A space \(X\) is compact if every open cover has a finite subcover: for any \(\{U_i\}\) with \(X \subseteq \bigcup U_i\), there exists a finite subset covering \(X\).

Examples

\([0, 1]\) is compact in \(\mathbb{R}\), but \((0, 1)\) is not, as the cover \(\{(1/n, 1)\}_{n=2}^\infty\) has no finite subcover.

Properties

Compact spaces are closed and bounded in \(\mathbb{R}^n\) (Heine-Borel), and continuous images of compact spaces are compact.