Complete uniform space

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)