# Compact space  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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.

## 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

${\mathcal {U}}=\{A_{\gamma }\mid A_{\gamma }\subset X,\,\gamma \in \Gamma \},$ where $\Gamma$ is an arbitrary index set, and satisfies

$A\subset \bigcup _{\gamma \in \Gamma }A_{\gamma }.$ An open cover is a cover in which all of the sets $A_{\gamma }$ are open. Finally, a subcover of ${\mathcal {U}}$ is a subset ${\mathcal {U}}'\subset {\mathcal {U}}$ of the form

${\mathcal {U}}'=\{A_{\gamma }\mid A_{\gamma }\subset X,\,\gamma \in \Gamma '\}$ with $\Gamma '\subset \Gamma$ such that

$A\subset \bigcup _{\gamma \in \Gamma '}A_{\gamma }.$ ## 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 $\Gamma '$ 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 $\{F_{\lambda }:\lambda \in \Lambda \}$ is a family of closed sets with empty intersection, $\bigcap _{\lambda \in \Lambda }F_{\lambda }=\emptyset$ , then a finite subfamily $\{F_{\lambda _{i}}:i=1,\ldots ,n\}$ has empty intersection $\bigcap _{i=1}^{n}F_{\lambda _{i}}=\emptyset$ .