Let 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 ≃ *ϱ for each ϱ ∈ Γ and each p ∈ X . We call an S-space if the relations ≃ and ≈ coincide on fin. S -spaces are interesting because their nonstandard hulls have representations within Nelson's (...) internal set theory . 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. (shrink)

Let ( * X, * T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in * X is defined as m(x) = μ T (st(x)). Consider the relation $R_{\mathrm{ns}} = \{\langle x, y \rangle \mid x, y \in \mathrm{ns} (^\ast X) \text{and} y \in m(x)\}.$ Frank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of R ns to the whole of * X. Wattenberg's (...) extensions of R ns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of R ns to * X should be possible in any topological space. A study of such extensions of R ns is the purpose of the present paper. We call a binary relation $W \subseteq ^\ast X \times ^\ast X$ an infinitesimal on * X if it is monadic and reflexive on * X. We prove, among other things, that the existence of an infinitesimal on * X that extends R ns is equivalent to the condition that the space (X, T) be regular. (shrink)