Please note the change of place and time in comparison to the previous semesters.

******************************************************************

A space \(X \subseteq 2^\omega\) is universally meager if for any Polish space \(Y\) and any continuous nowhere constant map \(f:Y \rightarrow 2^\omega\) the preimage \(f^{-1}[X]\) is meager in \(Y\). We present a forcing property \((\dagger)\), which is a strengthening of properness and implies that no dominating reals are added. It is known that many classical forcing posets like Cohen, Sacks and Miller satisfy this property. We showed that property \((\dagger)\) is preserved by countable support iterations.

We use this preservation result to prove that if we have such an iteration of length \(\omega_2\) over a model of CH, where the single forcings have size at most \(\omega_1\), all universally meager sets \(X \subseteq 2^\omega\) have size at most \(\omega_1\) in the forcing extension.

This has multiple set-theoretic applications:

• There is no perfectly meager space of size continuum in the Miller model.
• There is no strong measure zero set of size continuum in the Miller model.
• It is consistent with ZFC that the Rothberger property coincides with being Hurewicz without perfect subsets.
• Miller proved in 2005 that there exists a strong measure zero set of size \(\omega_1\) iff there exists a Rothberger space of size \(\omega_1\). We get that it is consistent with ZFC to have a strong measure zero set of size \(\omega_2\), but no Rothberger spaces of size \(\omega_2\).

This is joint work with Piotr Szewczak (University of Warsaw) and Lyubomyr Zdomskyy (TU Wien).