Weakly Separated Spaces and Pixley–Roy Hyperspaces

In this paper we obtain new results regarding the chain conditions in the Pixley–Roy hyperspaces 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 then we show that 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 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 F[X] is weakly Lindelöf and X satisfies any of the following conditions: