Nonstandard topologies with bases that consist only of standard sets

Abstract:

Let $X$ be an infinite set, $D$ a set of pseudo-metrics on $X,$ $\Xi \subset \ ^*X,$ and $\Gamma \subset \ ^*D.$ If $\rho (a,b)$ is limited (finite) for every $a,b\in \Xi$ and every $\rho \in \Gamma ,$ then, for each $\rho \in \Gamma ,$ we can define a pseudo-metric $\tilde \rho$ on $\Xi$ by writing $\tilde \rho (a,b)=$st$(\rho (a,b)).$ We investigate the conditions under which the topology induced on $\Xi$ by $\{\tilde \rho : \ \rho \in \Gamma \}$ has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder’s theorem, Mazur’s theorem, and Gelfand-Philips’s theorem.