On uniform spaces with invariant nonstandard hulls

Abstract

\begin{abstract} {\footnotesize Let$\langle X,\Gamma\rangle$   be a uniform space with its uniformity generated by a set of pseudo-metrics  $\Gamma$.  Let thesymbol $``\simeq"$  denote the usual infinitesimal relation on $^*X$, and define a new infinitesimal relation$``\approx"$  by writing  $x\approx y$ whenever  $ ^*\rho(x,p)\simeq\ ^*\rho(y,p)$  for each \ $\rho\in\Gamma$ \ and each \ $p\in X$. We show that a uniform space  $\langle X,\Gamma\rangle$  has invariant nonstandard hulls if and only if the relations  $\simeq$  and$\approx$  coincide on  $\text{fin}(^*X)$.  We provide several internal and external formulations of the latter condition and use them to explore further properties of uniform spaces that satisfy these conditions. We discuss some interesting examples, hereditary properties,  and their relations with topological spaces which admit only one compatible uniformity. Applications of the theory developed in this paper include a simple nonstandard proof of the interesting result that, in a regular Lindel\"{o}f space,  a subset is relatively compact if and only if it is pseudo-compact. \\[.15in] \bigskipKey Words: Uniform spaces, Nonstandard Topology}\end{abstract}