kompaktset

Kompaktset, or compact set, is a concept in topology describing a subset K of a topological space X with a specific covering property: every open cover of K has a finite subcover. This definition captures a sense of finiteness for infinite sets and underpins many results in analysis and geometry.

In metric spaces, compactness has several equivalent formulations. A subset K is compact if and only if every sequence in K has a convergent subsequence whose limit lies in K (sequential compactness). Equivalently, K is complete and totally bounded. In Euclidean spaces, the Heine-Borel theorem provides a concrete criterion: a set is compact iff it is closed and bounded.

Further properties include stability under continuous mappings. If f: X → Y is continuous and K ⊆ X is compact, then the image f(K) is compact in Y. Conversely, the preimage of a compact set under a continuous map is compact. In Hausdorff spaces, compact subsets are closed, which connects compactness to the separation of points by neighborhoods.

Common examples include closed intervals [a, b], the unit circle, the Cantor set, and any finite set. Non-examples include the open interval (0, 1) and the whole real line R, which are not compact in their standard topologies.

Kompaktset plays a central role in analysis and topology because many problems become tractable when restricting to compact domains. For instance, continuous functions on compact sets attain maximum and minimum values, and compactness often ensures the existence of convergent subsequences, enabling robust convergence arguments.