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. 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.

via Complete metric space – Wikipedia, the free encyclopedia.

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