Weakly Separated Spaces and Pixley–Roy Hyperspaces

Abstract

AbstractIn this paper we obtain new results regarding the chain conditions in the Pixley–Roy hyperspaces $${\mathscr {F}\hspace{0mm}}[X]$$ F [ X ] . For example, if c(X) and R(X) denote the cellularity and weak separation number of X (see Sect. 4) and we define the cardinals $$\begin{aligned} c^* (X):= \sup \{c(X^{n}): n\in {\mathbb {N}}\} \quad \text {and} \quad R^{*}(X):= \sup \{R(X^{n}): n\in {\mathbb {N}}\}, \end{aligned}$$ c ∗ ( X ) : = sup { c ( X n ) : n ∈ N } and R ∗ ( X ) : = sup { R ( X n ) : n ∈ N } , then we show that $$R^{*}(X) = c^ {*}\left( {\mathscr {F}\hspace{0mm}}[X]\right) $$ R ∗ ( X ) = c ∗ F [ X ] . On the other hand, in Sakai (Topol Appl 159:3080–3088, 2012, Question 3.23, p. 3087) Sakai asked whether the fact that $${\mathscr {F}\hspace{0mm}}[X]$$ 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 $${\mathscr {F}\hspace{0mm}}[X]$$ F [ X ] is weakly Lindelöf and X satisfies any of the following conditions: X is a Hausdorff k-space; X is a countably tight $$T_1$$ T 1 -space; X is weakly separated, then X is hereditarily separable.