In this paper, we prove that if a space X has a point-countable c n -network, then the Pixley-Roy hyperspace P R X also has a point-countable c n -network. If X is a regular space with a point-countable c k -network, then so does the Pixley-Roy hyperspace P R X . Moreover, if X has a point-countable s p -network (resp., strict Pytkeev network), then the Pixley–Roy hyperspace P R 2 X also has a point-countable s p -network (resp., strict Pytkeev network). On the other hand, we show that if the Pixley–Roy hyperspace P R X has a countable c n -network (resp., s p -network and strict Pytkeev network), then so does X . By these results, we obtain that if the Pixley–Roy hyperspace P R X is a cosmic space (resp., P 0 -space, strict P 0 -space, and stric P 0 -space), then so is X . Furthermore, the Pixley-Roy hyperspace P R n S 2 is not a k -space for each n ≥ 2 .
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