On Baire property, compactness and completeness properties of spaces of Baire functions
Alexander V. Osipov
TL;DR
The paper resolves the Banakh-Gabriyelyan problem for all $0\le\alpha\le\omega_1$ with Frechet targets by showing that the Baire property of $B_{\alpha}(X,\mathbb{R})$ governs the Baire behavior of $B_{\alpha}(X,Y)$ for any Frechet $Y$, and that when $\alpha\ge 2$, $B_{\alpha}(X,Y)$ is always Baire regardless of $X$. It develops a general equivalence framework across Frechet targets using almost-open mappings and projection/extension arguments, unifying Baire properties across function spaces. The study proceeds to demonstrate a broad array of completeness and compactness equivalences in $B_{\alpha}(X,Y)$, showing that several internal properties (e.g., $G_{\delta}$-density, $\omega$-fullness, countable base-compactness, and (strong) Choquet) are equivalent and often implied by Baire-ness. These results imply robust stability of Baire-like and Choquet-type properties in spaces of Baire functions, and they provide practical criteria linking the structural properties of $X$ (such as discreteness conditions) to the completeness of $B_{\alpha}(X,Y)$ for $\alpha=0,1$. The findings unify several strands of topological function space theory and have potential implications for analysis on spaces of Baire functions.
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 $α$ be a countable ordinal. Characterize topological spaces $X$ and $Y$ for which the function space $B_α(X,Y)$ is Baire. In this paper, for any Frechet space $Y$ , we have obtained a characterization topological spaces $X$ for which the function space $B_α(X,Y)$ is Baire. In particular, we proved that $B_α(X,\mathbb{R})$ is Baire if and only if $B_α(X,Y)$ is Baire for any Banach space $Y$. Also we proved that many completeness and compactness properties coincide in spaces $B_α(X,Y)$ for any Frechet space $Y$.