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| − | ''Cauchy sequence, of points in a metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422401.png" />''
| + | #REDIRECT[[Cauchy sequence]] |
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| − | A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422403.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422404.png" /> there is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422405.png" /> such that, for all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422407.png" />.
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| − | A generalization of a Cauchy sequence is the concept of a generalized Cauchy sequence (cf. [[Generalized sequence|Generalized sequence]]) in a uniform space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422408.png" /> be a uniform space with uniformity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f0422409.png" />. A generalized sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224012.png" /> is a directed set, is called a generalized Cauchy sequence if for every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224013.png" /> there is an index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224014.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224015.png" /> that come after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042240/f04224018.png" />.
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| − | ====References====
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.L. Kelley, "General topology" , Springer (1975)</TD></TR></table>
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| − | ====Comments====
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| − | 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)|Net (of sets in a topological space)]].)
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Latest revision as of 09:50, 9 December 2013
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