compact space

Per the compactness criteria for Euclidean space as stated in the Heine–Borel theorem, the interval A = (−∞, −2] is not compact because it is not bounded. The interval C = (2, 4) is not compact because it is not closed (but bounded). The interval B = [0, 1] is compact because it is both closed and bounded. In mathematics, especially general topology and 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 compactness: a topological space is compact if every open cover has a finite subcover. In metric spaces this is equivalent to several other formulations, including sequential compactness, though these equivalences can fail in more general topological spaces. Thus every sequence in the closed unit interval [0,1] has a convergent subsequence with limit in [0,1], whereas this fails for spaces such as the open interval (0,1) and the real line. For subsets of Euclidean space, compactness is equivalent to being closed and bounded, by the Heine–Borel theorem. The property of compactness often allows local information to be combined into global conclusions. The term compact set may refer either to a compact topological space or, more commonly, to a subset of a topological space that is compact in the subspace topology.