In this paper, we study the relation between a space $X$ satisfying certain generalized metric properties and the Pixley-Roy hyperspace $\texttt{PR}[X]$ over $X$ satisfying the same properties. We prove that if $\texttt{PR}[X]$ is starcompact (resp., star-Lindelöf), then $X$ is compact (resp., Lindelöf). 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 starcompact spaces. Moreover, we show that $\texttt{PR}[X]$ is strongly starcompact (resp., strongly star-Lindelöf) if and only if $X$ is finite (resp., countable). By these results, we obtain that $\texttt{PR}[X]$ is set strongly starcompact (resp., cosmic, set strongly star-Lindelöf) if and only if $X$ is finite (resp., countable).