# Compact space

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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

${\displaystyle {\mathcal {U}}=\{A_{\gamma }\mid A_{\gamma }\subset X,\,\gamma \in \Gamma \},}$

where ${\displaystyle \Gamma }$ is an arbitrary index set, and satisfies

${\displaystyle A\subset \bigcup _{\gamma \in \Gamma }A_{\gamma }.}$

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

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

with ${\displaystyle \Gamma '\subset \Gamma }$ such that

${\displaystyle A\subset \bigcup _{\gamma \in \Gamma '}A_{\gamma }.}$

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