In some metric space, not every Cauchy sequence have a limit (e.g.
While in some other metric spaces such as
A metric space
Complete metric spaces are intrinsically closed: no matter what space one places them in, they are always closed sets. More precisely:
- Let
be a metric space, and let be a subspace of . If is complete then must be closed under . - Conversely, suppose
is a closed subset of . Then the subspace is complete if is complete.