Ideals, Well-Orderable Families, and Baire Category in Truss's Feferman-Type Model

Abstract

Let $\mathsf{BCT}_{\mathsf{WO}}$ assert that every well-orderable family of dense open subsets of a perfect Polish space has dense intersection, and let $\mathsf{RS}_{\mathsf{WO}}$ assert that for every nonempty countable partial order, there is a filter meeting every member of any given well-orderable family of dense subsets. Over $\mathsf{ZF}$ they are shown to be equivalent. We show that in Truss's model, they characterize those $A\subseteq2^ω$ for which every $A$-indexed family of dense open sets has dense intersection. Assuming choice for well-orderable families of nonempty sets, we obtain an analogous characterization for total relations $R\subseteq X\times Y$ with meager vertical sections, whenever $Y$ is a surjective image of $2^ω$. The Feferman-type model $\mathfrak N_{\aleph_1}$ studied by Truss \cite{Truss1974} satisfies these hypotheses and $\mathsf{DC}$. We study the ideal of well-orderable subsets of $2^ω$ and its relations to other ideals in this model. We show, among other things: the well-orderable subsets of $2^ω$ are exactly the sets which are Rothberger in every finite power, have strong measure zero, are universally null, are Marczewski null, or contain no perfect subset. The ideal of well-orderable subsets of $2^ω$ is closed under well-ordered unions and is incomparable with the meager ideal. The equivalence with the Rothberger property does not extend to the Hurewicz property. Further more, in this model every set is Marczewski measurable, arbitrary maps into separable metric spaces have continuous perfect restrictions, and every set of positive outer measure contains a perfect subset.