complete metric space

(redirected from Cauchy complete)

complete metric space

[kəm′plēt ¦me·trik ′spās]
(mathematics)
A metric space in which every Cauchy sequence converges to a point of the space. Also known as complete space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

complete metric space

(theory)
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].
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References in periodicals archive ?
A Q-category A is said to be Cauchy complete when each left adjoint distributor with codomain A is represented by a functor [Lawvere, 1973], that is, when for each Q-category B the functor in (1) determines an equivalence of ordered sets
This clearly implies that the functor in (1) restricts to a biequivalence of locally ordered 2-categories between [Cat.sub.cc] (Q), the full subcategory of Cat(Q) determined by the Cauchy complete Q-categories, and Map(Dist(Q)).
Therefore the quantaloid Dist(Q) is equivalent to its full subquantaloid [Dist.sub.cc] (Q) whose objects are the Cauchy complete Q-categories.
Importantly, a result of [Heymans and Stubbe, 2011] says that, if Q is a Cauchy-bilateral quantaloid, then the symmetric-completion and the Cauchy-completion of any symmetric Q-category coincide, and the symmetrisation of a Cauchy complete Q-category is symmetrically complete.
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).
Therefore we may still suppose that A and B are Cauchy complete. Referring to the above, the category R is clearly symmetric whenever A and B are, and the left adjoint distributors represented by the functors S: R[right arrow]B and T: R[right arrow] A are evidently symmetric left adjoints.
In [Heymans and Stubbe, 2012] we showed that, for a Cauchy-bilateral quantaloid Q, every symmetric and Cauchy complete Q-category is a discrete object of [Cat.sub.cc](Q).
Proposition 3.16 If Q is a small quantaloid of closed cribles, then a Cauchy complete Q-category is discrete in [Cat.sub.cc](Q) if and only if it is symmetric.
But, because A is Cauchy complete, for any such span (f, g) we can consider the tensors x [cross product] f and y [cross product] g in A, and writing U = dom(f) = dom(g) we indeed have