Compact space

In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).

Related Topics:
Mathematics - Euclidean space - Closed - Bounded - Unit interval - Integer

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A more modern approach is to call a topological space compact if all its open covers have a finite subcover. The Heine-Borel theorem affirms that this coincides with "closed and bounded" for subsets of Euclidean space.

Related Topics:
Topological space - Heine-Borel theorem

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Note: Some authors such as Bourbaki use the term "quasi-compact" instead and reserve the name "compact" for topological spaces that are Hausdorff and compact.

Related Topics:
Bourbaki - Hausdorff

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