Heine–Borel theorem

In mathematical analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements are equivalent:S is closed and boundedevery open cover of S has a finite subcover, that is, S is compact.In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the Heine–Borel theorem for arbitrary metric spaces reads:A subset of a metric space is compact if and only if it is complete and totally bounded.
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