The
theorem of Heine-Borel 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.