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'' $  X $''

A separated [[Complete uniform space|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>