Complete metric space

A complete metric space is a metric space $(X,d)$ in which every Cauchy sequence converges (as a convergent sequence) to a point of $X$ .

Remarks

The sequence of rational decimal truncations of $\sqrt 2$ is Cauchy in $\mathbb{Q}$ , but it has no limit in $\mathbb{Q}$ . Completeness therefore depends on the space as well as on its metric.

Completeness is central in analysis and topology; for example it interacts strongly with compactness (see compact iff complete and totally bounded) and with the Baire category theorem (see complete metric spaces are Baire).

Examples

$(\mathbb{R}^n,\|\cdot\|_2)$ is complete.
$(\mathbb{Q},|\cdot|)$ is not complete.