compact space

Compactness is the topological substitute for “finite”. A finite set is wonderfully tame — any function on it attains a maximum, any cover can be trimmed to a few sets — and compactness manufactures those same conveniences for infinite spaces. The closed interval [0, 1], though it has uncountably many points, behaves in many ways like a finite set, while the open interval (0, 1) does not.

The definition uses covers. A space X is compact if every open cover of X has a finite subcover: whenever a family of open sets together cover X, finitely many of them already cover X. This “you can always get by with finitely many” condition is deceptively strong and is the modern, topology-only definition of compactness.

Compactness has spectacular consequences. A continuous real-valued function on a compact space attains its maximum and minimum (the extreme value theorem); the continuous image of a compact space is compact; a compact subset of a Hausdorff space is closed; and a continuous bijection from a compact space to a Hausdorff space is automatically a homeomorphism. Honesty demands a caveat: in general topological spaces compactness and sequential compactness are different notions, agreeing only under extra hypotheses such as metrizability.