Nonstandard topologies with bases that consist only of standard sets

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.