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Fundamental sequence

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Cauchy sequence, of points in a metric space

A sequence , such that for any there is a number such that, for all numbers , .

A generalization of a Cauchy sequence is the concept of a generalized Cauchy sequence (cf. Generalized sequence) in a uniform space. Let be a uniform space with uniformity . A generalized sequence , , where is a directed set, is called a generalized Cauchy sequence if for every element there is an index such that for all that come after in , .

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3] J.L. Kelley, "General topology" , Springer (1975)


Comments

Since generalized sequences are also called nets, one also speaks of Cauchy nets in uniform spaces. (Cf. also Net (of sets in a topological space).)

How to Cite This Entry:
Fundamental sequence. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fundamental_sequence&oldid=11357
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article