Menger and Consonant Sets in the Sacks Model

By a space we mean a metrizable separable zero-dimensional space.  A space X is Menger if for any sequence U_0, U_1,... of open covers of X, there are finite families F_0 subset U_0, F_1 subset U_1,... such that the family Union_{n in omega} F}_n covers X. If, moreover, the F_n's can be chosen in such a way that for every x in X, x is in Union F_n holds for all but finitely many n, X is said to be Hurewicz.  We call a space totally imperfect if it contains no copy of 2^omega.  We shall discuss how using Sacks combinatorics due to Miller with the Menger game yields that there are no totally imperfect Menger sets of reals of size continuum, c, in the Sacks model. Therefore, the Menger property behaves in the Sacks model as an instance of the Perfect Set Property, sets are either small or contain a perfect set. For models, which satisfy that d =c, there is always a totally imperfect Menger set of size continuum. (There are also models with small dominating number, where such sets exist.) Thus, combined with our result the existence of totally imperfect Menger sets of reals of size c is independent from ZFC.

Consonant spaces were introduced by Dolecki, Greco and Lechicki in 1995 and for the case X a subset of  2^\omega characterized by Jordan in 2020 using a topological game on the complement 2^omega - X. Motivated by a conjecture of Gartside, Medini and Zdomskyy about the structure of the Tukey order for hyperspaces of compact subspaces, we consider a grouped version of the Menger game and use a similar approach like for the Menger space result to conclude that every consonant and every Hurewicz subspace of 2^omega, as well as their complements, can be written as the union of omega_1-many compact sets in the Sacks model. In particular, there are only continuum many consonant spaces and Hurewicz spaces in this model.

This is joint work with Piotr Szewczak (Cardinal Stefan Wyszy\'nski University in Warsaw) and Lyubomyr Zdomskyy (TU Vienna).