Topological properties of spaces of Baire functions

A fundamental result proved by Bourgain, Fremlin and Talagrand states that the space B₁(M) of Baire one functions over a Polish space M is an angelic space. Stegall extended this result by showing that the class B₁(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_α(X,G) of Baire-α functions, where α 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_α(X,G) is a κ-Fréchet–Urysohn space and hence it is an Ascoli space, and (2) B_α(X,G) is a k-space iff X is countable.