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 \(B_1(X)\) of all Baire-one real-valued functions is the Banakh–Gabriyelyan problem of characterization of a topological space X for which the function space \(B_1(X)\) is Baire. In this paper, we solve this problem, namely, we obtain a characterization when \(B_1(X)\) is Baire for a topological space X. Also we prove that \(B_1(X)\) is Baire for any \(\gamma \) -space X and obtain a characterization of a topological space X for which \(B_1(X)\) is a Choquet space. This answers questions posed recently by Taras Banakh and Saak Gabriyelyan. We also conclude that it is consistent with ZFC that there is no uncountable separable metrizable space X such that \(B_1(X)\) is countable dense homogeneous.
