Menger and consonant sets in the Sacks model

TL;DR

This paper investigates the properties of Menger, Hurewicz, and consonant sets within the Sacks model, demonstrating independence results and analyzing their structure using iterated Sacks forcing and topological games.

Contribution

It establishes the independence of the existence of a totally imperfect Menger set of continuum size from ZFC and analyzes the structure of Hurewicz and consonant sets in the Sacks model.

Findings

Existence of a totally imperfect Menger set is independent of ZFC.

Structural analysis of Hurewicz and consonant sets in the Sacks model.

Use of topological games and iterated Sacks forcing in proofs.

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.