The Fréchet–Urysohn property of Pixley–Roy hyperspaces

Let F[X] be the Pixley–Roy hyperspace of a regular space X. In this paper, we prove the following theorem.Theorem – For a space X, the following are equivalent:(1)F[X] is a k-space;(2)F[X] is sequential;(3)F[X] is Fréchet–Urysohn;(4)Every finite power of X is Fréchet–Urysohn for finite sets;(5)Every finite power of F[X] is Fréchet–Urysohn for finite sets.As an application, we improve a metrization theorem on F[X].