Topological properties of spaces of Baire functions

TL;DR

This paper investigates the topological properties of spaces of Baire-$oldsymbol{ extalpha}$ functions, extending known results for Baire one functions to higher classes and more general target groups, revealing their Fréchet–Urysohn and $k$-space characteristics.

Contribution

It generalizes the topological analysis of Baire function spaces to higher classes and non-precompact groups, establishing new properties like $oldsymbol{ extkappa}$-Fréchet–Urysohn and conditions for $k$-spaces.

Findings

$B_oldsymbol{ extalpha}(X,G)$ is a $oldsymbol{ extkappa}$-Fréchet–Urysohn space.

$B_oldsymbol{ extalpha}(X,G)$ is a $k$-space iff $X$ is countable.

Spaces are Ascoli spaces due to their $oldsymbol{ extkappa}$-Fréchet–Urysohn property.

Abstract

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.