The $Δ_1$-property of $X$ is equivalent to the Choquet property of $B_1(X)$
Alexander V. Osipov
TL;DR
The paper characterizes the $Δ_1$-property of a Tychonoff space $X$ via the Choquet property of the function space $B_1(X)$ of Baire-one functions with the pointwise topology, proving $Δ_1$-property ⇔ $B_1(X)$ is Choquet (and related pseudocomplete and disjointness conditions). It further shows that for almost $K$-analytic spaces, these conditions are equivalent to $C_p(X)$ being $ ext{κ}$-Fréchet-Urysohn, $B_1(X)$ being Baire, and $X$ having the $( ext{κ})$ and $Δ_1$ properties, with $X$ also being $k$-scattered. In addition, the authors provide ZFC constructions—an example of a separable pseudocompact space $X$ with $C_p(X)$ $ ext{κ}$-Fréchet-Urysohn but not $Δ_1$ and a dense subspace $P$ with $B_1(P)$ Baire but not Choquet—addressing open questions of Kakol–Leiderman–Tkachuk. These results deepen the understanding of the $B_1(X)$-based Choquet framework in $C_p(X)$-theory and yield concrete counterexamples and equivalences with implications for selection principles and topological games.
Abstract
We give a characterization of the $Δ_1$-property of any Tychonoff space $X$ in terms of the function space $B_1(X)$ of all Baire-one real-valued functions on a space $X$ with the topology of pointwise convergence. We establish that for a Tychonoff space $X$ the $Δ_1$-property is equivalent to the Choquet property of $B_1(X)$. Also we construct under $ZFC$ an example of a separable pseudocompact space $X$ such that $C_p(X)$ is $κ$-Frechet-Urysohn but $X$ fails to be a $Δ_1$-space. This answers a question of Kakol-Leiderman-Tkachuk.