Abstract
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 ] .
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