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We consider the hyperspace $CL\left(X\right)$ of nonempty closed subsets of completely metrizable space $X$ endowed with the Wijsman topologies ${\tau }_{W_d}$. If $X$ is separable and $d$, $e$ are two metrics generating the topology of $X$, every countable set closed in $\left(CL\left(X\right),{\tau }_{W_e}\right)$ has isolated points in $\left(CL\left(X\right),{\tau }_{W_d}\right)$. For $d=e$ , this implies a theorem of Costantini on topological completeness of $\left(CL\left(X\right),{\tau }_{W_d}\right)$. We show that for nonseparable $X$ the hyperspace $\left(CL\left(X\right),{\tau }_{W_d}\right)$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.