compactness

Compactness is a central concept in topology and analysis describing a form of finiteness for spaces or subsets. A subset K of a topological space X is compact if every open cover of K has a finite subcover; that is, from any collection of open sets whose union contains K, one can select a finite number of them that still cover K. Equivalently, every open cover has a finite subcover. In metric spaces, compactness is also characterized by sequential properties: every sequence in K has a convergent subsequence whose limit lies in K.

In metric spaces, compactness is further equivalent to being complete and totally bounded. In Euclidean spaces, the Heine-Borel theorem gives a concrete criterion: a subset is compact if and only if it is closed and bounded.

Key properties include: continuous images of compact sets are compact; finite unions of compact sets are compact; closed subsets of compact spaces are compact; and in Hausdorff spaces, compact sets are closed. Every infinite subset of a compact space has a limit point within that space, and every sequence in a compact space has a convergent subsequence.

Local variations include local compactness, where every point has a compact neighborhood, and the concept of compactifications, such as the one-point compactification of non-compact spaces.

Examples help illustrate the notion. In R with the usual metric, the closed interval [0,1] is compact, while (0,1) is not. In infinite-dimensional normed spaces, closed and bounded sets need not be compact; for instance, the closed unit ball is not compact in infinite dimensions, contrasting with finite-dimensional spaces where the unit ball is compact by Heine-Borel.

Applications of compactness include the extreme value theorem (continuous functions achieve maxima and minima on compact sets) and the guarantee of uniform continuity on compact domains.