Compact space

A compact topological space is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental topological properties in analysis and topology.

Intuitively, compactness can be understood as a generalization of being "closed and bounded". In Euclidean spaces , by the Heine–Borel theorem, a set is compact if and only if it is closed and bounded.

However, in a general topological space, a metric is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In , this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.

Definition

A topological space is compact if for every collection of open sets in such that

there exists a finite subcollection such that

See also

• topological properties
• Heine-Borel theorem

Terminology

en	fr	de	zh		ja
compact space	espace compact	kompakter Raum	紧空间	緊空間	コンパクト空間
compact	compact	kompakt	紧的	緊的	コンパクト的
compactness	compacité	Kompaktheit	紧性	緊性	コンパクト性