# 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.