Complete metric space

In mathematics, completeness is a property ascribed to 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."

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 associated 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.
• The rational numbers Q are not complete. For example, the sequence (x_n) defined by x₀ = 1, x_n+1 = 1 + 1/x_n is Cauchy, but does not converge in Q.

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 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 G_δ, that is, a G_δ in every topological space in which it can be embedded.

See also

• Banach space
• Hilbert space