Weakly separated spaces and Pixley-Roy hyperspaces

In this paper we obtain new results regarding the chain conditions in the Pixley-Roy hyperspaces $\mathcal{F}[X]$. For example, if $c(X)$ and $R(X)$ denote the cellularity and weak separation number of $X$ (see Section~[4]) and we define the cardinals $$c^{\ast }(X):=\sup \{c(X^n):n\in \mathbb{N}\}\text{and}{R}^{\ast }(X):=\sup \{R(X^n):n\in \mathbb{N}\},$$ then we show that $R^{\ast }(X)=c^{\ast }(\mathcal{F}[X])$. On the other hand, in “M. Sakai, Cardinal functions of Pixley-Roy hyperspaces, Topol. Appl., 159 (2012), 3080–3088.” Sakai asked whether the fact that $\mathcal{F}[X]$ is weakly Lindelöf implies that $X$ is hereditarily separable and proved that if $X$ is countably tight then the previous question has an affirmative answer. We shall expand Sakai’s result by proving that if $\mathcal{F}[X]$ is weakly Lindelöf and $X$ is a Hausdorff $k$-space; or $X$ is a countably tight $T_1$ space; or $X$ is weakly separated, then $X$ is hereditarily separable.