# Complete metric space – Wikipedia, the free encyclopedia

Complete metric space

In mathematical analysis, a metric space M is called complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M.

Intuitively, a space is complete if there are no “points missing” from it (inside or at the boundary). Thus, a complete metric space is analogous to a closed set. For instance, the set of rational numbers is not complete, because e.g. $\sqrt{2}$ is “missing” from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to “fill all the holes”, leading to the completion of a given space, as explained below.