# Complete metric space

In mathematics, a **complete metric space** is a metric space in which every Cauchy sequence in that space is *convergent*. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete."

## Contents

## Formal definition

Let *X* be a metric space with metric *d*. Then *X* is complete if for every Cauchy sequence <math>x_1,x_2,\ldots \in X</math> there is an associated element <math>x \in X</math> such that <math>\mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0</math>.

## Examples

- The real numbers
**R**, and more generally finite-dimensional Euclidean spaces, with the usual metric are complete. - Any compact metric space is sequentially compact and hence complete. The converse does not hold: for example,
**R**is complete but not compact. - In a space with the discrete metric, the only Cauchy sequences are those which are constant from some point on. Hence any discrete metric space is complete.
- The rational numbers
**Q**are*not*complete. For example, the sequence (*x*_{n}) defined by*x*_{0}= 1,*x*_{n+1}= 1 + 1/*x*_{n}is Cauchy, but does not converge in**Q**.

## Completion

Every metric space *X* has a **completion** <math>\bar X</math> which is a complete metric space in which *X* is isometrically embedded as a dense subspace. The completion has a universal property.

### Examples

- The real numbers
**R**are the completion of the rational numbers**Q**with respect to the usual metric of absolute distance.

## Topologically complete space

Completeness is not a topological property: it is possible for a complete metric space to be homeomorphic to a metric space which is not complete. For example, the map

- <math> t \leftrightarrow \left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right) </math>

is a homeomorphism between the complete metric space **R** and the incomplete space which is the unit circle in the Euclidean plane with the point (0,-1) deleted. The latter space is not complete as the non-Cauchy sequence corresponding to *t*=*n* as *n* runs through the positive integers is mapped to a non-convergent Cauchy sequence on the circle.

We can define a topological space to be *metrically topologically complete* if it is homeomorphic to a complete metric space. A topological condition for this property is that the space be metrizable and an *absolute G _{δ}*, that is, a G

_{δ}in every topological space in which it can be embedded.