# Completion of a uniform space

A separated complete uniform space for which there exists a uniformly-continuous mapping such that for any uniformly-continuous mapping from into a separated complete uniform space there exists a unique uniformly-continuous mapping with . The subspace is dense in and the image of entourages in under are entourages in ; their closures in constitute a fundamental system of entourages in . If is separated, then is injective (this allows one to identify with ). The separated completion of a subspace is isomorphic to the closure of . The separated completion of a product of uniform spaces is isomorphic to the product of the separated completions of the spaces occurring as factors in the product.

The proof of the existence of generalizes in fact Cantor's construction of the set of real numbers from the set of rational numbers.

#### References

[1] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |

**How to Cite This Entry:**

Completion of a uniform space. M.I. Voitsekhovskii (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Completion_of_a_uniform_space&oldid=15398