Completely Baire spaces, Menger spaces, and projective sets

Franklin D. Tall; Lyubomyr Zdomskyy

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arXiv · 1803.03458

Completely Baire spaces, Menger spaces, and projective sets

Franklin D. Tall·Lyubomyr Zdomskyy

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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.

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BibTeX RIS

Franklin D. Tall, Lyubomyr Zdomskyy. 2018-03-09. Completely Baire spaces, Menger spaces, and projective sets. https://arxiv.org/abs/1803.03458

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