Topological properties of spaces of Baire functions

A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space $B_1(M)$ of Baire one functions over a Polish space $M$ is an angelic space. Stegall extended this result by showing that the class $B_1(M,E)$ of Baire one functions valued in a normed space $E$ is angelic. These results motivate our study of various topological properties in the classes $B_\alpha(X,G)$ of Baire-$\alpha$ functions, where $\alpha$ is a nonzero countable ordinal, $G$ is a metrizable non-precompact abelian group and $X$ is a $G$-Tychonoff first countable space. In particular, we show that (1) $B_\alpha(X,G)$ is a $\kappa$-Fr\'{e}chet--Urysohn space and hence it is an Ascoli space, and (2) $B_\alpha(X,G)$ is a $k$-space iff $X$ is countable.