Compact space
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.
Examples
Some examples of compact spaces:
- The closed unit interval [0,1].
- The n-sphere, for every natural number n.
- Any finite topological space.
- The Cantor set.
- The Stone-Čech compactification of any Tychonoff space.
Theorems
Some theorems related to compactness (see the Topology Glossary for the definitions):
- A closed subset of a compact space is compact.
- A compact subset of a Hausdorff space is closed.
- A nonempty compact subset of the real numbers has a greatest element and a least element.
- A subset of Euclidean n-space is compact if and only if it is closed and bounded. (Heine-Borel theorem)
- A metric space is compact if and only if it is complete and totally bounded.
- A metric space is compact if and only if every sequence in the space has a subsequence with limit in the space.
- A continuous image of a compact space is compact.
- The product of any collection of compact spaces is compact. (Tychonoff's Theorem -- this is equivalent to the axiom of choice)
- A compact Hausdorff space is normal.
- Every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.
- A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.
- A topological space is compact if and only if every net on the space has a subnet which has a limit in the space.
- 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)
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.)
- Pseudocompact: Every real-valued continuous function on the space is bounded.
- 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.
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