compacité

Compacité, or compactness, is a central notion in topology and analysis describing a form of finiteness that persists under many operations. A subset K of a topological space X is compact if every open cover of K has a finite subcover. In intuitive terms, a compact set can be approached with only finitely many basic pieces, regardless of how large K is.

In metric spaces there are several equivalent characterizations. One is sequential compactness: every sequence in K has a subsequence that converges to a limit in K. In Euclidean space, the Heine-Borel theorem provides a practical criterion: a subset of R^n is compact if and only if it is closed and bounded. More generally, a subset of a metric space is compact if and only if it is complete and totally bounded.

Important consequences and properties include: the continuous image of a compact set is compact; compact sets are closed in Hausdorff spaces; finite unions of compact sets are compact; and every open cover of a compact set admits a finite subcover. Compactness is preserved under continuous mappings and is stable under taking closed subsets of compact sets.

In product spaces, compactness can be more delicate. Tychonoff’s theorem states that any product of compact spaces is compact (assuming the axiom of choice), a foundational result in topology with broad implications.

Examples and non-examples help illustrate the concept. The closed interval [0,1] is compact in R, while the open interval (0,1) is not. A finite set is compact, as are spheres and other closed and bounded sets in R^n. Non-compact sets include the whole real line R or open intervals like (0,1).

Compactness provides a unifying framework for convergence, continuity, and finiteness across many areas of mathematics.