2021/10/29 by Osipov, Alexander V.
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2110.15496
A topological space X is Baire if the Baire Category Theorem holds for X, i.e., the intersection of any sequence of open dense subsets of X is dense in X. One of the interesting problems for the space B1(X) of all Baire-one real-valued functions is characterization topological space X for which the function space B1(X) is Baire. In this paper, we solve this problem, namely, we have obtained a characterization when a function space B1(X) has the Baire property for any Tychonoff space X. Also we proved that B1(X) is Baire for any γ-space X. This answers a question posed recently by T. Banakh and S. Gabriyelyan. We also conclude that, it is consistent there are no uncountable separable metrizable space X such that B1(X) is countable dense homogeneous.