In some metric space, not every Cauchy sequence have a limit (e.g. ).

While in some other metric spaces such as , every Cauchy sequences converge. This dichotomy motivates the following definition:

A metric space is said to be complete iff every Cauchy sequence in is convergent in .

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.