Let (X,𝒰) be a uniform space
in some set theoretical structure ℳ and let ∗x be the set corresponding to X in an
enlargement ∗ℳ of ℳ In this paper a set of 𝒰-finite elements of ∗X is defined and
this set is used to define a nonstandard hull of (X,𝒰). The main result is that, with
some specific exceptions depending on the existence of measurable cardinal numbers,
this nonstandard hull is the same as the smallest of the nonstandard hulls defined by
Luxemburg. This result is used in giving a characterization of subsets of X on
which every uniformly continuous, real valued function is bounded. Also, two
examples are given to illustrate the possible structure of the nonstandard
hulls.
