kompaktisuus

Compactness, or kompaktisuus, is a fundamental concept in topology and analysis that captures a strong form of finiteness. In a topological space, compactness means that every open cover of the space has a finite subcover. Equivalently, one can express this property through finite intersection behavior of closed sets or through sequential criteria in metric spaces.

In metric spaces, compactness is equivalent to sequential compactness: every sequence has a convergent subsequence. It is also equivalent to being complete and totally bounded. In Euclidean spaces, the Heine–Borel theorem provides a concrete characterization: a subset of R^n is compact if and only if it is closed and bounded. Classic examples include a closed interval [a, b], the unit circle, and finite unions of compact sets. By contrast, open intervals and unbounded sets are not compact.

Several important properties follow from compactness. The continuous image of a compact set is compact, and in Hausdorff spaces, compact sets are closed. The finite intersection property offers another perspective: a space is compact if every collection of closed sets with the finite intersection property has a nonempty intersection. In product spaces, the product of any family of compact spaces is compact (the general statement is a form of the Tychonoff theorem, which relies on the axiom of choice).

Beyond topology, compactness appears in other areas of mathematics. In mathematical logic, the compactness theorem states that if every finite subset of a set of first-order sentences has a model, then the whole set has a model, illustrating a related but distinct notion of compactness.