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# Compact space

### From Citizendium, the Citizens' Compendium

In mathematics, a **compact space** is a topological space for which every covering of that space by a collection of open sets has a finite subcovering. If the space is a metric space then compactness is equivalent to the set being complete and totally bounded and again equivalent to sequential compactness: that every sequence in the set has a convergent subsequence.

A subset of a topological space is compact if it is compact with respect to the subspace topology. A compact subset of a Hausdorff space is closed, but the converse does not hold in general. For the special case that the set is a subset of a finite dimensional normed space, such as the Euclidean spaces, then compactness is equivalent to that set being closed and bounded: this is the Heine-Borel theorem.

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## Cover and subcover of a set

Let *A* be a subset of a set *X*. A **cover** for *A* is any collection of subsets of *X* whose union contains *A*. In other words, a cover is of the form

where is an arbitrary index set, and satisfies

An **open cover** is a cover in which all of the sets are open. Finally, a **subcover** of is a subset of the form

with such that

## Formal definition of compact space

A topological space *X* is said to be **compact** if *every* open cover of *X* has a *finite* subcover, that is, a subcover which contains at most a finite number of subsets of *X* (in other words, the index set is finite).

## Finite intersection property

Just as the topology on a topological space may be defined in terms of the closed sets rather than the open sets, so we may transpose the definition of compactness in terms of open sets into a definition in terms of closed sets. A space is compact if the closed sets have the *finite intersection property*: if is a family of closed sets with empty intersection, , then a finite subfamily has empty intersection .

## Examples

- Any finite space.
- An indiscrete space.
- A space with the cofinite topology.
- The
*Heine-Borel theorem*: In Euclidean space with the usual topology, a subset is compact if and only if it is closed and bounded.

## Properties

- Compactness is a topological invariant: that is, a topological space homeomorphic to a compact space is again compact.
- A closed set in a compact space is again compact.
- A subset of a Hausdorff space which is compact (with the subspace topology) is closed.
- The quotient topology on an image of a compact space is compact
- The image of a compact space under a continuous map to a Hausdorff space is compact.
- The Cartesian product of two (and hence finitely many) compact spaces with the product topology is compact.
- The
*Tychonoff product theorem*: The product of any family of compact spaces with the product topology is compact. This is equivalent to the Axiom of Choice. - If a space is both compact and Hausdorff then no finer topology on the space is compact, and no coarser topology is Hausdorff.