compact

A topological space $X$ is if, for every collection $\{U_{i}\}_{i\in I}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection $\{U_{i_{j}}\}_{j=1}^{n}$ 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 Hausdorff 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