Some remarks on Pixley-Roy hyperspaces - Luong, Quoc Tuyen; Ong, Van Tuyen; Nguyen, Xuan Truc (Hue University Journal of Science: Natural Science, 2025)
In this paper, we study cellular-compact, cellular-Lindel"of, strongly star-Hurewicz, strongly star-Rothberger, strongly star-Menger spaces on hyperspaces with the Pixley-Roy topology. For a space X and nN, we prove that (1) If PR[X] or PR_n[X] is cellular-compact, then X is cellular-compact. However, there exists a compact space X such that |X|=ω, but PR_n[X] for all nN and PR[X] are not cellular-compact spaces. (2) If PR[X] or PR_n[X] is cellular-Lindel"of, then X is cellular-Lindel"of. (3) X is countable if and only if PR[X] is a strongly star-Hurewicz space, if and only if PR[X] is a strongly star-Rothberger space if and only ifPR[X] is a strongly star-Menger space.