Theorem of Heine-Borel

The theorem of Heine-Borel in analysis states:

A subset of the real numbers is compact if and only if it is closed and bounded.

This is true not only for the real numbers, but also for some other metric spaces: the complex numbers, the p-adic numbers, and Euclidean space. However, it fails for the rational numbers and for infinite dimensional normed vector spaces.

The theorem is closely related to the theorem of Bolzano-Weierstrass.

The proper generalization is:

A subset of a metric space is compact if and only if it is complete and totally bounded.

The terms complete and totally bounded are explained in the metric space article.