Cellular-compact and cellular-Lindelöf on hyperspaces

The generalized metric properties on hyperspaces with the Pixley-Roy topology and the Vietoris topology have been studied by many authors. They considered several generalized metric properties and studied the relation between a space $X$ satisfying such property and its hyperspaces with the Pixley-Roy topology and the Vietoris topology satisfying the same property. In this paper, we study cellular-compact and cellular-Lindelöf spaces on hyperspaces with the Pixley-Roy topology and the Vietoris topology. For a space $X$, we prove that $\texttt{PR}[X]$ is cellular-compact (resp., cellular-Lindelöf) if and only if $X$ is finite (resp., countable). Moreover, if $\mathbb K(X)$ or $\mathcal F(X)$ or $\mathcal F_n(X)$ ($ n\in\mathbb N$) is cellular-compact (resp., cellular-Lindelöf), then $X$ is cellular-compact (resp., cellular-Lindelöf).