Geodesic
Note on the Wijsman hyperspaces of completely metrizable spaces
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 827-832.

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We consider the hyperspace $CL(X)$ 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 $(CL(X), \tau_{W_{e}})$ has isolated points in $(CL(X), \tau_{W_{d}})$. For $d=e$ , this implies a theorem of Costantini on topological completeness of $(CL(X), \tau_{W_{d}})$. We show that for nonseparable $X$ the hyperspace $(CL(X), \tau_{W_{d}})$ may contain a closed copy of the rationals. This answers a question of Zsilinszky.
Consideriamo sugli spazi $CL(X)$ dei sottoinsiemi chiusi e non vuoti di uno spazio $X$ completamente metrizzabile la topologia di Wijsman $\tau_{W_{d}}$. Se $X$ è separabile, mostriamo che, per ogni metrica $d$, $e$ su $X$, ogni insieme chiuso e numerabile in $(CL(X), \tau_{W_{e}})$ ha punti isolati in $(CL(X), \tau_{W_{d}})$. Se $d=e$ , questo implica il teorema di Costantini sulla completezza topologica di $(CL(X), \tau_{W_{d}})$. Per $X$ non-separabili, rispondiamo ad una questione sollevata da Zsilinszky, mostrando che in molti casi gli spazi $(CL(X), \tau_{W_{d}})$ contengono copie chiuse dei razionali. 
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Chaber, J.; Pol, R. Note on the Wijsman hyperspaces of completely metrizable spaces. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 827-832. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_3_a15/

[1] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. | MR | Zbl

[2] C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc., 123 (1995), 2569-2574. | MR | Zbl

[3] C. Costantini, On the hyperspace of a non-separable metric space, Proc. Amer. Math. Soc., 126 (1998), 3393-3396. | MR | Zbl

[4] E. Van Douwen, The integers in topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.) North Holland, Amsterdam 1984, 116-167. | MR | Zbl

[5] R. Engelking- Mrówka, On $E$-compact spaces, Bull. Acad. Pol. Sci., 6 (1958), pp. 429-439. | MR | Zbl

[6] A. S. Kechris, Classical Descriptive Set Theory, Springer Verlag, New York, 1994. | MR | Zbl

[7] L. Zsilinszky, Polishness of the Wijsman topology revisited, Proc. Amer. Math. Soc., 126 (1998), pp. 3763-3765. | MR | Zbl