On uniform spaces with invariant nonstandard hulls

Let 〈X,Γ〉 be a uniform space with its uniformity generated by a set of pseudo-metrics Γ . Let the symbol “ ' " denote the usual infinitesimal relation on ∗X , and define a new infinitesimal relation “ ≈ " by writing x ≈ y whenever ∗ρ(x, p) ' ∗ρ(y, p) for each ρ ∈ Γ and each p ∈ X . Let us call a uniform space 〈X,Γ〉 an S-space if the relations ' and ≈ coincide on fin(∗X). In [14], we showed that every S-space whose uniformity is generated by a single pseudometric has invariant nonstandard hulls. Here we extend that result to all uniform spaces and use it to explore further properties of S-spaces, which can now be recognized as uniform spaces that have invariant nonstandard hulls. 2010 Mathematics Subject Classification 54J05 (primary); 54D20, 54D60 (secondary)