On the Baire Property of the Space of Baire Indicator Functions

A topological space $$X$$ has the Baire property if any countable intersection of open dense subsets of $$X$$ is dense in $$X$$ . An interesting problem in the theory of function spaces is that of characterizing the Baire property of a function space in terms of a topological property of the supports of functions. A solution of this problem for the space $$B_{\alpha}(X,\{0,1\})$$ of Baire class $$\alpha$$ indicator functions, where $$1\leq \alpha\leq \omega_1$$ , is presented. Namely, given a Tychonoff space $$X$$ , a criterion for the space $$B_{\alpha}(X,\{0,1\})$$ to have the Baire property in terms of a topological property of $$X$$ is obtained. This result answers a question of T. Banakh and S. Gabriyelyan in the class of spaces of Baire indicator functions.