Compact space

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In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle. For subsets of Euclidean space, the analogous statement is sequential compactness: a set is compact if and only if every infinite sequence in the set has a subsequence that converges to a point of the set. Likewise, whereas every real-valued function on a finite set is bounded and attains its maximum and minimum, every continuous real-valued function on a compact space has these properties. For compact subsets of Euclidean space, this is the extreme value theorem. Another basic property of finite sets is that every cover of a finite set by subsets has a finite subcover: one may choose, for each point of the finite set, a member of the cover containing it. The corresponding topological property is used to define...

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