Complete metric space

In mathematics, a complete metric space is a metric space in which every Cauchy sequence 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."

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

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. Thus, some bounded complete metric spaces are not compact.
 * The rational numbers Q are not complete. For example, the sequence (xn) defined by x0 = 1, xn+1 = 1 + 1/xn is Cauchy, but does not converge in Q. (In R it converges to an irrational number.)

Completion
Every metric space X has a completion $$\bar X$$ 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 real line R is homeomorphic to an open interval, say, (0,1). Another example: the map


 * $$ t \leftrightarrow \left(\frac{2t}{1+t^2},\frac{1-t^2}{1+t^2}\right) $$

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 Gδ, that is, a Gδ in every topological space in which it can be embedded (or just Gδ in its completion in a chosen metric). In particular, all open subsets of Euclidean spaces are metrically topologically complete.