Universally meager sets in the Miller model and similar ones

We work in the realm of sets of reals. We prove that in the Miller model and in a model constructed by Goldstern-Judah-Shelah all universally meager sets have size at most $ω_1$. Some relations between combinatorial covering properties in these models allow to obtain the same limitations for sizes of Rothberger spaces and Hurewicz spaces with no homeomorphic copy of the Cantor set inside. It follows from our results that the existence of a strong measure zero set of size $ω_2$ does not imply the existence of a Rothberger space of size $ω_2$. We also prove that in the Miller model all strong measure zero sets have size at most $ω_1$.

💡 Research Summary

The paper investigates the size of several “small” subsets of the Cantor space 2^ω in two specific forcing extensions of the universe of set theory: the Miller model and the model constructed by Goldstern, Judah, and Shelah (GJS model). The main objects of study are universally meager (UM) sets, Rothberger spaces, Hurewicz spaces, and strong measure zero (SMZ) sets.

A central technical tool is the property (†) introduced in