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. As a topological space, a subset 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.