Some Results on Pixley–Roy Hyperspaces

In this paper, we prove that if a space X has a point‐countable cn‐network, then the Pixley‐Roy hyperspace PR[X] also has a point‐countable cn‐network. If X is a regular space with a point‐countable ck‐network, then so does the Pixley‐Roy hyperspace PR[X]. Moreover, if X has a point‐countable sp‐network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace PR2[X] also has a point‐countable sp‐network (resp., strict Pytkeev network). On the other hand, we show that if the Pixley–Roy hyperspace PR[X] has a countable cn‐network (resp., sp‐network and strict Pytkeev network), then so does X. By these results, we obtain that if the Pixley–Roy hyperspace PR[X] is a cosmic space (resp., ‐space, strict ‐space, and stric ‐space), then so is X. Furthermore, the Pixley‐Roy hyperspace PRn[S2] is not a k‐space for each n ≥ 2.