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Compact set

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In mathematics, a compact set is a set for which every covering of that set by a collection of open sets has a finite subcovering. If the set is a subset of a metric space then compactness is equivalent to the set being complete and totally bounded or, equivalently, that every sequence in the set has a convergent subsequence. 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.

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 Γ 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 Aγ 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 set

A subset A of a set X is said to be compact if every open cover of A 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 Γ' is finite).

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