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

<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>.