Selective separability of Pixley–Roy hyperspaces

A space X is said to be selectively separable (=M-separable) if for every sequence { D n : n ∈ ω } of dense subsets of X, there are finite sets F n ⊂ D n ( n ∈ ω ) such that ⋃ { F n : n ∈ ω } is dense in X. We show that the Pixley–Roy hyperspace PR ( X ) of a space X is selectively separable if and only if X is countable and every finite power of X has countable fan-tightness for finite sets. As an application, under b = d there are selectively separable Pixley–Roy hyperspaces PR ( X ) , PR ( Y ) such that PR ( X ) × PR ( Y ) is not selectively separable.