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Compact space

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In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is "small" in a certain sense and "contains all its limit points". The modern general definition calls a topological space compact if every open cover of it 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 use the term 'quasicompact' instead and reserve the term 'compact' for compact Hausdorff spaces, but Wikipedia follows the usual current practice of allowing compact spaces to be non-Hausdorff.

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Equivalent definitions of a compact set in Rn

For any subset of Euclidean space Rn, the following three conditions are equivalent:

  • Every open cover has a finite subcover. This is the definiton most commonly used, as stated above.
  • Every sequence in the set has a convergent subsequence.
  • The set is closed and bounded. This is the condition that is easiest to verify, for example a closed interval or closed n-ball.

In other spaces, these conditions may or may not be equivalent, depending on the properties of the space.

Examples of compact spaces

Theorems

Some theorems related to compactness (see the Topology Glossary for the definitions):

  • A continuous image of a compact space is compact.

  • A closed subset of a compact space is compact.

  • A nonempty compact subset of the real numbers has a greatest element and a least element.

  • A compact Hausdorff space is normal.

  • A metric space is compact if and only if every sequence in the space has a subsequence with limit in the 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.

  • A topological space can be embedded in a compact Hausdorff space if and only if it is a Tychonoff space.

  • A metric space X is compact if and only if every metric space homeomorphic to X is complete.

  • If the metric space X is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (Lebesgue's number lemma)

  • 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)

  • Two compact Hausdorff spaces X1 and X2 are homeomorphic if and only if their rings of continuous real-valued functions C(X1) and C(X2) are isomorphic.

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.



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