HeineBorel

Heine-Borel refers to a fundamental result in real analysis that characterizes compact subsets of Euclidean space. The theorem is named after Eduard Heine and Émile Borel and is central to the study of topology and analysis in finite-dimensional spaces.

In the n-dimensional Euclidean space R^n, a subset is compact if and only if it is closed and bounded. Equivalently, every open cover of the set has a finite subcover. The forward implication holds for any compact set; the converse is specific to R^n (and more generally to finite-dimensional normed spaces), where closed and bounded sets must be compact under any norm.

The Heine-Borel theorem has several key consequences. On compact sets, continuous functions are bounded and attain their maximum and minimum values (the extreme value theorem). In metric spaces, compactness can also be characterized by sequential compactness: every sequence has a convergent subsequence. These ideas underpin many results in analysis, such as integration, optimization, and the study of convergence.

Generalizations and related concepts include the notion of a Heine-Borel space: a metric space in which every closed and bounded subset is compact. This property holds in Euclidean spaces but fails in infinite-dimensional spaces, where closed and bounded sets need not be compact. In such settings, compactness is replaced by total boundedness together with completeness.

History-wise, the result is traditionally attributed to Heine and Borel, with early work in the late 19th and early 20th centuries. It is a cornerstone of the interplay between topology and analysis in finite-dimensional spaces.