Compactness Basics

Compactness is a core concept in topology, generalizing the idea of boundedness in Euclidean spaces. It ensures a space can be covered by a finite number of open sets, making it a powerful tool in analysis and geometry. This guide from MathMultiverse explores the definition, properties, and examples of compactness, providing a clear foundation for understanding this essential topological property.

Definition

A topological space \(X\) is compact if every open cover of \(X\) has a finite subcover. Formally, for any collection of open sets \(\{U_i\}\) such that \(X \subseteq \bigcup U_i\), there exists a finite subset of indices such that \(X \subseteq \bigcup_{i=1}^n U_i\).

\[ X \text{ is compact} \iff \forall \{U_i\} \text{ open with } X \subseteq \bigcup U_i, \exists \text{ finite } \{U_{i_1}, \ldots, U_{i_n}\} \text{ with } X \subseteq \bigcup_{k=1}^n U_{i_k} \]

Examples

Consider the interval \([0, 1]\) in \(\mathbb{R}\). It is compact by the Heine-Borel theorem, as it is closed and bounded. However, the open interval \((0, 1)\) is not compact, as the open cover \(\{(1/n, 1)\}_{n=2}^\infty\) has no finite subcover.

\[ \bigcup_{n=2}^\infty (1/n, 1) = (0, 1) \]

Properties

Compact spaces in \(\mathbb{R}^n\) are closed and bounded (Heine-Borel theorem). Additionally, continuous functions from a compact space are bounded and attain their bounds, and their images are compact.

\[ f: X \to Y \text{ continuous}, X \text{ compact} \implies f(X) \text{ is compact} \]