A remark on uniform spaces with invariant nonstandard hulls

✍ Scribed by Nader Vakil; Roozbeh Vakil

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 94 KB
• Volume: 51
• Category: Article
• ISSN: 0044-3050
• DOI: 10.1002/malq.200510011

✦ Synopsis

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 * (x, p) * (y, p) for each ∈ 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.