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.