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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

\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.



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