Menger and consonant sets in the Sacks model

TL;DR

The paper addresses the independence of the existence of a totally imperfect Menger set of cardinality $\mathfrak{c}$ in the Cantor cube by employing iterated Sacks forcing to create a model with $\mathfrak{d}<\mathfrak{c}$ and analyzing the Menger, Hurewicz, and consonant properties via game-theoretic characterizations. It develops a suite of topological games (Menger, grouped Menger, weak grouped Menger) and forcing-combinatorics tools to connect the behavior of reals under Sacks forcing with the size and structure of Menger-type sets; it also studies Hechler forcing to illustrate that a rich family of Hurewicz and Rothberger subspaces can coexist with $\mathfrak{d}<\mathfrak{c}$. The main results show that in the Sacks model, totally imperfect Menger sets have size at most $\omega_1$, while consonant and Hurewicz sets, and their complements, decompose into $\omega_1$ many compact subspaces; the paper concludes with several open problems about the precise boundaries between Menger, S1(Gamma,O), and consonant properties. Overall, the work advances understanding of how combinatorial set theory and topological selection principles interact under forcing to shape the landscape of small sets in $2^{\omega}$.

Abstract

Using iterated Sacks forcing and topological games, we prove that the existence of a totally imperfect Menger set in the Cantor cube with cardinality continuum is independent from ZFC. We also analyze the structure of Hurewicz and consonant subsets of the Cantor cube in the Sacks model.

Figures (1)

• Figure :

Theorems & Definitions (44)

• Theorem 1.1: Bartoszyński, Tsaban BaTs
• Remark 1.3
• Theorem 1.4: Hurewicz
• Theorem 1.5: Scheepers coc1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.3: Miller mill
• Lemma 3.5: Miller mill