A topological spaceMathworldPlanetmath X is compactPlanetmathPlanetmath if, for every collectionMathworldPlanetmath {Ui}iI of open sets in X whose union is X, there exists a finite subcollection {Uij}j=1n whose union is also X.

A subset Y of a topological space X is said to be compact if Y with its subspace topology is a compact topological space.

Note: Some authors require that a compact topological space be HausdorffPlanetmathPlanetmath as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. The modern convention seems to be to use compact in the sense given here, but the old definition is still occasionally encountered (particularly in the French school).

Title compact
Canonical name Compact
Date of creation 2013-03-22 11:53:35
Last modified on 2013-03-22 11:53:35
Owner djao (24)
Last modified by djao (24)
Numerical id 11
Author djao (24)
Entry type Definition
Classification msc 54D30
Classification msc 81-00
Classification msc 83-00
Classification msc 82-00
Classification msc 46L05
Classification msc 22A22
Related topic QuasiCompact
Related topic LocallyCompact
Related topic HeineBorelTheorem
Related topic TychonoffsTheorem
Related topic Compactification
Related topic SequentiallyCompact
Related topic Lindelof
Related topic NoetherianTopologicalSpace
Defines compact set
Defines compact subset