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Note on the Wijsman hyperspaces of completely metrizable spaces. (English) Zbl 1098.54006

The authors consider the hyperspace \(CL(X)\) of nonempty closed subsets of a completely metrizable space \(X\) endowed with the Wijsman topology \(\tau_{W_d}\). When \(X\) is separable and \(d\) and \(e\) are two metrics generating the topology of \(X\) then every countable set closed in \((CL(X),\,\tau_{W_e})\) has isolated points in \((CL(X),\,\tau_{W_d})\). In the case \(d=e\) this statement implies a theorem of Costantini on topological completeness of \((CL(X),\,\tau_{W_d})\). In this paper it is shown that for a nonseparable space \(X\) the hyperspace \((CL(X),\,\tau_{W_d})\) may contain a closed copy of the rationals. This answers a question of Zsilinszky.

MSC:

54B20 Hyperspaces in general topology
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