Heine–Borel theorem

The Heine-Borel Theorem in analysis states:

A subset of the real numbers R 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 Rⁿ. 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 to arbitrary metric spaces is:

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