The weak Hurewicz property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. We show that F[X] is weakly Hurewicz in the sense of Kočinac if and only if X is countable. This answers a question in Kočinac [6]. Moreover, making use of the ideas in Daniels [2], we redefine the weak Hurewicz property, and show that (1) if F[X] is weakly Hurewicz, then every finite power of X is Hurewicz, (2) conversely if X is semi-stratifiable and every finite power of X is Hurewicz, then F[X]λ is weakly Hurewicz for any cardinal λ.