Completely Baire spaces, Menger spaces, and projective sets
Tall, Franklin D. ; Zdomskyy, Lyubomyr, 1983-
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 ▫$\boldsymbol{V=L}$▫ for projective counterexamples, and the Axiom of Projective Determinacy for positive results. For the first problem, the first author, S. Todorcevic, and S. Tokgöz 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 ▫$\omega^{\rm th}$▫ power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.
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