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. m space X, the following are equivalent:(1) ] is a k-space; ] is sequential; ] is Fréchet–Urysohn; finite power of X is Fréchet–Urysohn for finite sets; finite power of F [ X ] is Fréchet–Urysohn for finite sets. application, we improve a metrization theorem on F [ X ] .