[Home]Compact space

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A compact space is a topological space in which every open cover has a finite subcover. That is, any collection of open sets whose union is the whole space has a finite subcollection whose union is still the whole space. Some authors reserve the term 'compact' for compact Hausdorff spaces, but this article follows the usual current practice of allowing compact spaces to be non-Hausdorff.


Some examples of compact spaces:


Some theorems related to compactness:

Other forms of compactness

There are a number of topological properties which are equivalent to compactness in metric spaces, but are inequivalent in general topological spaces. These include the following.

While all these concepts are equivalent for metric spaces, in general we have the following implications: Compact spaces are countably compact. Sequentially compact spaces are countably compact. Countably compact spaces are pseudocompact and weakly countably compact.


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Last edited December 16, 2001 2:13 am by AxelBoldt (diff)