# Compact space

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.

## Contents

## 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

- <math>\mathcal{U}=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma \},</math>

where <math>\Gamma</math> is an arbitrary index set, and satisfies

- <math>A \subset \bigcup_{\gamma \in \Gamma }A_{\gamma}.</math>

An **open cover** is a cover in which all of the sets <math>A_\gamma</math> are open. Finally, a **subcover** of <math>\mathcal{U}</math> is a subset <math>\mathcal{U}' \subset \mathcal{U}</math> of the form

- <math>\mathcal{U}'=\{A_{\gamma} \mid A_{\gamma} \subset X,\,\gamma \in \Gamma'\}</math>

with <math>\Gamma' \subset \Gamma</math> such that

- <math>A \subset \bigcup_{\gamma \in \Gamma'}A_{\gamma}.</math>

## 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 <math>\Gamma'</math> 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 <math>\{ F_\lambda : \lambda \in \Lambda \}</math> is a family of closed sets with empty intersection, <math>\bigcap_{\lambda \in \Lambda} F_\lambda = \emptyset</math>, then a finite subfamily <math>\{ F_{\lambda_i} : i=1,\ldots,n \}</math> has empty intersection <math>\bigcap_{i=1}^n F_{\lambda_i} = \emptyset</math>.

## 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.