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 $\text{[math]}$ , 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 $\text{[math]}$ , 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 $\text{[math]}$ is compact if for every collection $\text{[math]}$ of opensets in $\text{[math]}$ such that

$\text{[math]}$

there exists a finite subcollection

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such that

$\text{[math]}$