On Baire property, compactness and completeness properties of spaces of Baire functions

TL;DR

This paper characterizes when spaces of Baire functions are Baire spaces, linking properties of the domain space with the Baire property of the function space, and explores related completeness and compactness properties.

Contribution

It provides a characterization of topological spaces X for which Baire function spaces B_α(X,Y) are Baire, especially for Frechet and Banach spaces, extending previous problems.

Findings

Baire property of B_α(X,Y) depends on the domain space X.

Baire property for real-valued Baire functions implies it for all Banach space-valued functions.

Various completeness and compactness properties coincide in B_α(X,Y) spaces.

Abstract

A topological space $X$ is Baire if the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems for the space of Baire functions is the Banakh-Gabriyelyan problem: Let $\alpha$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_{\alpha}(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_{\alpha}(X,Y)$ is Baire. In particular, we proved that $B_{\alpha}(X,\mathbb{R})$ is Baire if and only if $B_{\alpha}(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_{\alpha}(X,Y)$ for any Frechet space $Y$.