A remark 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 “ ≈ ” on ∗X by writing x ≈ y whenever ∗ρ(x, p) ' ∗ρ(y, p) for each ρ ∈ Γ and each p ∈ X. We call (X,Γ) an S-space if the relations ' and ≈ coincide on fin(∗X). S-spaces are interesting because their nonstandard hulls have representations within Nelson’s Internal Set Theory [5]. This was shown in [1], where it was also observed that the class of uniform spaces that have invariant nonstandard hulls is contained in the class of S-spaces. The question of whether there are S-spaces that do not have invariant nonstandard hulls was left open in [1]. In this note we show that when the uniformity of an S-space is given by a single pseudometric, the space has invariant nonstandard hulls.