Tömörségi

Tömörségi is the Hungarian term used to describe the mathematical concept of compactness. In topology, a subset K of a topological space X is compact if every open cover of K has a finite subcover; that is, from any collection of open sets whose union contains K, one can select finitely many that still cover K.

In metric spaces, compactness has several equivalent characterizations. A set is compact if and only if every sequence in the set has a subsequence that converges to a point within the set (sequential compactness). In metric spaces, compactness is also equivalent to being complete and totally bounded. In Euclidean spaces, the Heine–Borel theorem provides a practical criterion: a subset of Rn is compact precisely when it is closed and bounded.

Important properties of tömörségi include:

- Continuous images of compact sets are compact.

- In Hausdorff spaces, compact sets are closed.

- A space is compact if and only if every finite intersection of closed sets with the finite intersection property is nonempty.

- The extreme value theorem: a continuous function defined on a compact space attains its maximum and minimum.

- Finite products of compact spaces are compact; the product of arbitrary collections is compact in general, by Tychonoff’s theorem.

Common examples and non-examples help illustrate the idea. The closed interval [0,1] in R is compact, as are the unit circle and any finite set. The whole real line R is not compact, nor is the open interval (0,1). In many areas of analysis and geometry, compactness provides a powerful framework for guaranteeing the existence of extrema, convergent subsequences, and stability under continuous mappings.