The $\Delta_1$-property of $X$ is equivalent to the Choquet property of $B_1(X)$

Abstract

We give a characterization of the $\Delta_1$-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 $\Delta_1$-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 $\kappa$-Frechet-Urysohn but $X$ fails to be a $\Delta_1$-space. This answers a question of Kakol-Leiderman-Tkachuk.