Completely Baire spaces, Menger spaces, and projective sets

TL;DR

This paper explores the properties of projective sets related to Baire and Menger spaces, providing new results under weaker axioms and constructing a space with unique Baire properties, advancing understanding in descriptive set theory.

Contribution

It extends classical theorems about analytic and co-analytic sets to projective sets using weaker axioms and constructs a novel space with specific Baire properties in ZFC.

Findings

Equivalence of certain projective set problems under weaker axioms

Construction of a space with ω-th power completely Baire but no dense completely metrizable subspace

Answers a question of Eagle and Tall in Abstract Model Theory

Abstract

W. Hurewicz proved that analytic Menger sets of reals are $\sigma$-compact and that co-analytic completely Baire sets of reals are completely metrizable. It is natural to try to generalize these theorems to projective sets. This has previously been accomplished by $V = L$ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokg\"oz have produced a finer analysis with much weaker axioms. We produce a similar analysis for the second problem, showing the two problems are essentially equivalent. We also construct in ZFC a separable metrizable space with $\omega$-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.