Baire property of space of Baire-one functions

Abstract

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 characterization topological space $X$ for which the function space $B_1(X)$ is Baire. In this paper, we solve this problem, namely, we have obtained a characterization when a function space $B_1(X)$ has the Baire property for any Tychonoff space $X$. Also we proved that $B_1(X)$ is Baire for any $\gamma$-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 $B_1(X)$ is countable dense homogeneous.