The Cantor tree, the $\gamma$-property, and Baire function spaces

Abstract:

Let $X \subseteq {2^\omega }$ and $T \cup X$ be the Cantor tree over $X$. We show that ${C_k}(T \cup X)$ is a Baire space if and only if $X$ is a $\gamma$-set. We obtain from this result consistent examples of spaces $Y$ and $Z$ such that ${C_k}(Y)$ and ${C_k}(Z)$ are Baire spaces but ${C_k}(Y) \times {C_k}(Z)$ is not a Baire space. It also follows that there are consistent examples of locally compact nonparacompact spaces $Y$ such that ${C_k}(Y)$ is Baire but not weakly $\alpha$-favorable.