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:
- The closed unit interval [0,1].
- Any finite topological space.
- The Cantor set.
Some theorems related to compactness:
- A closed subset of a compact space is compact.
- A compact subset of a Hausdorff space is closed.
- A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine-Borel theorem)
- A nonempty compact subset of the real numbers has a greatest element and a least element.
- The product of any collection of compact spaces is compact. (Tychonoff's Theorem -- this is equivalent to the axiom of choice)
- A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
- A continuous image of a compact space is compact.
- A topological space is compact if and only if every filter on the space has a convergent refinement.
- A topological space is compact if and only if every ultrafilter on the space is convergent.
- If a topological space has a sub-base such that every cover of the space by members of the sub-base has a finite subcover, then the space is compact. (Alexander's Sub-base Theorem)
- A metric space is compact if and only if it is complete and totally bounded.
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.
- Sequentially compact: Every sequence has a convergent subsequence.
- Countably compact: Every countable open cover has a finite subcover. (Or, equivalently, every infinite subset has an ω-accumulation point.)
- Weakly countably compact: Every infinite subset has an accumulation point.
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.