precompactness

Precompactness is a topological property of a subset within a topological space. A subset K of a topological space X is said to be precompact if its closure in X is a compact subset. This means that every open cover of the closure of K admits a finite subcover.

In metric spaces, a set is precompact if and only if its closure is totally bounded. A set is totally bounded if for every positive real number epsilon, there exists a finite number of points in the set such that the distance between any point in the set and its nearest point in this finite collection is less than epsilon. This is equivalent to stating that for every epsilon > 0, the set can be covered by a finite number of open balls of radius epsilon.

The concept of precompactness is particularly useful when dealing with properties that are inherited by compact sets but are not necessarily preserved by taking closures. For instance, in functional analysis, precompact sets are important for proving the Arzelà-Ascoli theorem, which provides conditions for a family of functions to contain a uniformly convergent subsequence. A precompact set is also sometimes referred to as a relatively compact set. The distinction between precompactness and compactness lies in whether the set itself must be compact, or if its closure must be compact.