A remark on uniform spaces with invariant nonstandard hulls

Abstract

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 (IST, [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. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)