Kompaktnos

Kompaktnos is a mathematical term used to describe a fundamental property of a space or a subset within a topological space. In informal terms, a compact set behaves much like a finite set in many respects, allowing for strong control over limits and coverings.

The standard definition uses open covers: a subset K of a topological space X is compact if every collection of open sets that covers K contains a finite subcollection that also covers K. In metric spaces, compactness is equivalent to sequential compactness: every sequence in K has a subsequence that converges to a point in K. In Euclidean spaces, the Heine–Borel theorem provides a practical criterion: a subset of R^n is compact if and only if it is closed and bounded.

Examples include the closed interval [0,1], the unit circle, and any finite set. The open interval (0,1) is not compact in R. The set {1/n : n in N} is not compact in R because it does not contain its limit point 0.

Key properties include that continuous images of compact sets are compact, finite unions of compact sets are compact, and closed subsets of a compact set are compact. Moreover, every continuous function defined on a compact space attains its maximum and minimum when the codomain is a real-valued space.

Compactness is central to existence proofs and analysis, providing a bridge between finiteness and infinity. The term appears in several languages with close variants, reflecting its foundational role in topology and analysis.