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 PRX 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 PRX. Moreover, if X has a point-countable sp-network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace PR2X also has a point-countable sp-network (resp., strict Pytkeev network). On the other hand, we show that if the Pixley–Roy hyperspace PRX 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 PRX is a cosmic space (resp., P0-space, strict P0-space, and stric P0-space), then so is X. Furthermore, the Pixley-Roy hyperspace PRnS2 is not a k-space for each n≥2.