Difference between revisions of "Complete uniform space"
From Encyclopedia of Mathematics
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| + | A [[Uniform space|uniform space]] in which every [[Cauchy filter|Cauchy filter]] converges. An important example is a [[Complete metric space|complete metric space]]. A closed subspace of a complete uniform space is complete; a complete subspace of a separable uniform space is closed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete. Any uniform space $X$ can be uniformly and continuously mapped onto some dense subspace of a complete uniform space $\hat{X}$ (see [[Completion of a uniform space|Completion of a uniform space]]). | ||
====References==== | ====References==== | ||
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| + | |valign="top"|{{Ref|Bo}}|| N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) | ||
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| + | |valign="top"|{{Ref|Is}}|| J.R. Isbell, "Uniform spaces" , Amer. Math. Soc. (1964) | ||
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| + | |valign="top"|{{Ref|Ke}}|| J.L. Kelley, "General topology" , Springer (1975) | ||
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| + | |} | ||
Latest revision as of 17:31, 9 December 2013
2020 Mathematics Subject Classification: Primary: 54E15 Secondary: 54E50 [MSN][ZBL]
A uniform space in which every Cauchy filter converges. An important example is a complete metric space. A closed subspace of a complete uniform space is complete; a complete subspace of a separable uniform space is closed. The product of complete uniform spaces is complete; conversely, if the product of non-empty uniform spaces is complete, then all the spaces are complete. Any uniform space $X$ can be uniformly and continuously mapped onto some dense subspace of a complete uniform space $\hat{X}$ (see Completion of a uniform space).
References
| [Bo] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |
| [Is] | J.R. Isbell, "Uniform spaces" , Amer. Math. Soc. (1964) |
| [Ke] | J.L. Kelley, "General topology" , Springer (1975) |
How to Cite This Entry:
Complete uniform space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_uniform_space&oldid=30895
Complete uniform space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_uniform_space&oldid=30895
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article