Complete metric space

2020 Mathematics Subject Classification: Primary: 54E50 [MSN][ZBL]

A metric space in which each Cauchy sequence converges. A complete metric space is a particular case of a complete uniform space. A closed subset $A$ of a complete metric $(X,d)$ space is itself a complete metric space (with the distance which is the restriction of $d$ to $A$). The converse is true in a general metric space: if $(X,d)$ is a metric space, not necessarily complete, and $A\subset X$ is such that $(A,d)$ is complete, then $A$ is necessarily a closed subset.

Given any metric space $(X,d)$ there exists a unique completion of $X$, that is a triple $(Y,\rho,i)$ such that:

• $(Y, \rho)$ is a complete metric space;
• $i: X \to Y$ is an isometric embedding, namely a map such that $d(x,y) = \rho (i(x), i(y))$ for any pair of points $x,y\in X$;
• $i(X)$ is dense in $Y$.

Often people refer to the metric space $(Y, \rho)$ as the completion. Both the space and the isometric embedding are unique up to isometries.

Comments

Completeness is not a topological property, that is, there are metric spaces which are homeomorphic as topological spaces, one being complete and the other not. For example, consider the real line $\mathbb{R}$ and the open unit interval $(-1,1)$, each with the usual metric. There are homeomorphic, for example via the map $x \mapsto x / (1 + |x|)$. However, as metric spaces, $\mathbb{R}$ is complete, but the sequence $1-1/n$ is a Cauchy sequence which does not converge in $(-1,1)$.

A compact metric space is complete.

A topological space is topologically complete if there is a complete metric space structure compatible with the given topology: this is a topological property.

References

[1]	Steen, Lynn Arthur; Seebach, J. Arthur Jr., Counterexamples in Topology (second edition). Springer-Verlag (1978). ISBN 978-0-486-68735-3 MR507446 Zbl 0386.54001