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Cauchy sequence

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2020 Mathematics Subject Classification: Primary: 40A05 Secondary: 54E35 [MSN][ZBL]

Cauchy sequence, of points in a metric space

A sequence \{x_i\} of elements in a metric space (X,d) such that for any \varepsilon > 0 there is a number N such that d (x_n, x_m) < \varepsilon \qquad \forall m,n\geq N\, . The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called complete, see also Cauchy criteria.

The concept of Cauchy sequence can be generalized to Cauchy nets (see also Net; Net (of sets in a topological space), Generalized sequence and Cauchy filter) in a uniform space. Let X be a uniform space with uniformity \mathcal{U}. A net \{x_\alpha, \alpha \in A\} of elements x_\alpha \in X (where A is a directed set) is called a Cauchy net if for every element U\in \mathcal{U} there is an index \alpha_0 \in A such that for all (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, .

References

[Al] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[Du] J. Dugundji, "Topology" , Allyn & Bacon (1966) MR0193606 Zbl 0144.21501
[Ke] J.L. Kelley, "General topology" , Springer (1975)
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Fundamental sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_sequence&oldid=30873
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article