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Complete uniform space

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2010 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://www.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