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 $n\in\mathbb N$, 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 $n\in\mathbb N$ 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 if$PR[X]$ is a strongly star-Menger space.