# Heine-Borel theorem

In mathematics, the **Heine-Borel theorem** characterises the compact subsets of the real numbers.

The real numbers form a metric space with the usual distance as metric. A subset of this metric space, treated as a topological space, is compact if and only if it is closed and bounded.

A Euclidean space of fixed finite dimension *n* also forms a metric space with the Euclidean distance as metric. As a topological space, the same statement holds: a subset is compact if and only if it is closed and bounded.

## Discussion

The theorem makes two assertions. Firstly, that a compact subset of **R** is closed and bounded. Proof (sketch): (a) a compact subset of any Hausdorff space is closed; (b) the metric is a continuous function, and a continuous function on a compact set is bounded.

The second and major part of the theorem is that a closed bounded subset of **R** is compact. We may reduce to the case of a closed interval, since a closed subset of a compact space is compact.

One proof in this case follows directly from the definition of compactness is terms of open covers. Consider an open cover of a closed interval [*a*,*b*]. Consider the subset *S* of [*a*,*b*] consisting of all *x* such that the interval [*a*,*x*] has a finite subcover. The set *S* is non-empty, since *a* is in *S*. If *b* were not in *S*, consider the supremum *s* of *S*, and show that there is another *t* between *s* and *b* which is also in *S*. This contradiction shows that *b* is in *S*, which establishes the result.

A second proof relies on the Bolzano-Weierstrass theorem to show that a closed interval is sequentially compact. This already shows that it is countably compact. But **R** is separable since the rational numbers **Q** form a countable dense set, and this applies to any interval as well. Hence countable compactness implies compactness.

Finally we note that a closed bounded subset of **R**^{n} is compact. Proof (sketch): a finite product of compact spaces is compact, and a closed bounded subset of **R**^{n} is a closed subset of a "closed box", that is, a finite product of closed bounded intervals.

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