2022/11/19 by Osipov, Alexander V.
#FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2211.10765
A topological space X is Baire if the intersection of any sequence of open dense subsets of X is dense in X. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let α be a countable ordinal. Characterize topological spaces X and Y for which the function space Bα(X,Y) is Baire. In this paper, for any Frechet space Y , we have obtained a characterization topological spaces X for which the function space Bα(X,Y) is Baire. In particular, we proved that Bα(X,ℝ) is Baire if and only if Bα(X,Y) is Baire for any Banach space Y. Also we proved that many completeness and compactness properties coincide in spaces Bα(X,Y) for any Frechet space Y.