2018/03/09 by Tall, Franklin D., Zdomskyy, Lyubomyr
#03C95 #03E15 #03E35 #03E60 #03F15 #54A35 #54D45 #54E50 #54F52 #54G20 #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · doi:10.48550/arxiv.1803.03458
W. Hurewicz proved that analytic Menger sets of reals are σ-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. 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 ω-th power completely Baire, yet lacking a dense completely metrizable subspace. This answers a question of Eagle and Tall in Abstract Model Theory.