Some function applications of weak $\lambda$-spaces

Abstract

In this paper it is proved that there exists (in $ZFC$) a separable pseudocompact weak $\lambda$-space $X$ which is not $\Delta_1$-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 $\gamma$-set $X$ which is not weak $\lambda$-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.