\text { Nonstandard } \chi \text {-hulls of uniform spaces }

Abstract

A Nonstandard hull of a metric space is constructed by working in a $\kappa$-saturated superstructure with $\kappa>\chi_0$. The hypothesis $\kappa>\chi_0$ accounts for the fact that we work with sequences in metric spaces. In uniform spaces, we naturally replace sequences by nets $\left(x_\alpha\right)_{\alpha \in D}$. Hence the need for focusing on the cardinality of $D$ arises here and hence the need for $\chi$-hulls for various cardinalities $\chi$. In this article we obtain a $\chi$-hull of a uniform space $X$ by considering $\kappa$-saturated superstructures $V\left({ }^* X\right)$ with $\kappa>\chi$.