Some remarks on Pixley-Roy hyperspaces

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 $\texttt{PR}[X]$ or $\texttt{PR}_n[X]$ is cellular-compact, then $X$ is cellular-compact. However, there exists a compact space $X$ such that $|X|=\omega$, but $\texttt{PR}_n[X]$ for all $n\in\mathbb N$ and $\texttt{PR}[X]$ are not cellular-compact spaces.(2) If $\texttt{PR}[X]$ or $\texttt{PR}_n[X]$ is cellular-Lindel"of, then $X$ is cellular-Lindel"of.(3) $X$ is countable if and only if $\texttt{PR}[X]$ is a strongly star-Hurewicz space, if and only if $\texttt{PR}[X]$ is a strongly star-Rothberger space if and only if$\texttt{PR}[X]$ is a strongly star-Menger space.