# Complete uniform space

From Encyclopedia of Mathematics

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

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article