Topological reduced products via good ultrafilters

Abstract

Good ultrafilters produce topological ultraproducts which enjoy a strong Baire category property (depending upon how good the ultrafilter is). We exploit this property to prove a “uniform boundedness” theorem as well as a theorem which says that, under the Generalized Continuum Hypothesis (GCH), many ultraproduct spaces have families consisting of closed discrete sets of high cardinality such that every nonempty open set contains one of these sets. In another section we relate the strong Baire properties to the infinite distributivity of Boolean Algebras of regular open sets. Finally, we prove that, under the GCH, a great many topological ultrapowers are homeomorphic to the corresponding ultrapower of the space of rational numbers; and we show further that the GCH is indispensable to the proof. A purely model-theoretic application of our methods solves a problem related to the Keisler-Shelah Ultrapower Theorem.