Completion of a uniform space

$ X $

A separated complete uniform space $ \widehat{X} $ for which there exists a uniformly-continuous mapping $ i : X \rightarrow \widehat{X} $ such that for any uniformly-continuous mapping $ f $ from $ X $ into a separated complete uniform space $ Y $ there exists a unique uniformly-continuous mapping $ g : \widehat{X} \rightarrow Y $ with $ f = g \circ i $. The subspace $ i ( X) $ is dense in $ \widehat{X} $ and the image of entourages in $ X $ under $ i \times i $ are entourages in $ i ( X) $; their closures in $ \widehat{X} \times \widehat{X} $ constitute a fundamental system of entourages in $ \widehat{X} $. If $ X $ is separated, then $ i $ is injective (this allows one to identify $ X $ with $ i ( X) $). The separated completion of a subspace $ A \subset X $ is isomorphic to the closure of $ i ( A) \subset \widehat{X} $. 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 $ \widehat{X} $ 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)