complete metric space

(redirected from Cauchy completion)

complete metric space

[kəm′plēt ¦me·trik ′spās]
A metric space in which every Cauchy sequence converges to a point of the space. Also known as complete space.

Complete Metric Space


a metric space in which the Cauchy convergence criterion is satisfied.

A sequence of points x1, x2, …, xn, … on a line, in a plane, or in space is said to be a Cauchy, or fundamental, sequence if for sufficiently large numbers n and m the distance between the points xn and xm becomes arbitrarily small. According to the Cauchy criterion, such a sequence of points has a limit if, and only if, it is a Cauchy sequence. For many sets of mathematical objects, such as functions or operators, a concept of distance can be introduced that has properties analogous to the properties of ordinary distance. The set is then said to be a metric space. The concept of the limit of a sequence of points in the metric space can be defined in the usual way. If the Cauchy criterion holds, the space is called complete.

Examples of complete metric spaces are Euclidean spaces and many other linear spaces—for example, the space of continuous functions on the interval [a, b] with distance

and Hubert space. A closed subset of a complete metric space is itself a complete metric space. If a metric space is not complete, it can be made complete in a way analogous to the way the set of rational numbers can be augmented by the irrationals to form the set of real numbers.

The concept of completeness can be extended to nonmetric topological spaces in which neighborhoods of different points can be compared. Examples of such spaces are topological groups and rings.

complete metric space

A metric space in which every sequence that converges in itself has a limit. For example, the space of real numbers is complete by Dedekind's axiom, whereas the space of rational numbers is not - e.g. the sequence a[0]=1; a[n_+1]:=a[n]/2+1/a[n].
References in periodicals archive ?
That is to say, each Q-category A has a Cauchy completion [A.
The Cauchy completion can of course be applied to a symmetric Q-category (assuming that Q is involutive), but the resulting Cauchy complete category need not be symmetric anymore: the functor [(-).
We can assume without loss of generality that A and B are Cauchy complete, because every Q-category is isomorphic to its Cauchy completion in Dist(Q).
If Q is Cauchy- biateral then the Cauchy completion of a symmetric Q -category is again symmetric, hence any symmetric Q-category is isomorphic to its Cauchy completion in SymDist(Q) (and not merely in Dist(Q)).
13); one particular consequence is that, for symmetric categories enriched in either quantaloid, the symmetric completion coincides with the Cauchy completion (cf.