# Complete uniform space

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 can be uniformly and continuously mapped onto some dense subspace of a complete uniform space (see Completion of a uniform space).

#### References

 [1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) [2] J.L. Kelley, "General topology" , Springer (1975)