Compact space

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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 sets of the form , where is an arbitrary index set, such that . For any such cover , a set of the form with and such that is said to be a subcover of .


Formal definition of compact set

A subset A of a set X is said to be compact if every cover of A has a finite subcover, that is, a subcover which contains at most a finite number of subsets of X (or has a finite index set).

See also

Open set

Closed set

Topological space