Some function applications of weak $λ$-spaces

Abstract

In this paper it is proved that there exists (in $ZFC$) a separable pseudocompact weak $λ$-space $X$ which is not $Δ_1$-space. It follows that there exists a space $X$ such that $B_1(X,[0, 1])$ is Choquet, while $B_1(X)$ is not Choquet. Also it is proved that there exists a $γ$-set $X$ which is not weak $λ$-set. It follows that there exists a zero-dimensional separable metrizable space $X$ such that $B_1(X)$ is Baire, while $B_1(X, [0,1])$ is not Choquet. These results answers the previously posed questions.