# Note on the Wijsman hyperspaces of completely metrizable spaces

Bollettino dell'Unione Matematica Italiana (2002)

- Volume: 5-B, Issue: 3, page 827-832
- ISSN: 0392-4041

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topChaber, J., and Pol, R.. "Note on the Wijsman hyperspaces of completely metrizable spaces." Bollettino dell'Unione Matematica Italiana 5-B.3 (2002): 827-832. <http://eudml.org/doc/196017>.

@article{Chaber2002,

abstract = {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.},

author = {Chaber, J., Pol, R.},

journal = {Bollettino dell'Unione Matematica Italiana},

language = {eng},

month = {10},

number = {3},

pages = {827-832},

publisher = {Unione Matematica Italiana},

title = {Note on the Wijsman hyperspaces of completely metrizable spaces},

url = {http://eudml.org/doc/196017},

volume = {5-B},

year = {2002},

}

TY - JOUR

AU - Chaber, J.

AU - Pol, R.

TI - Note on the Wijsman hyperspaces of completely metrizable spaces

JO - Bollettino dell'Unione Matematica Italiana

DA - 2002/10//

PB - Unione Matematica Italiana

VL - 5-B

IS - 3

SP - 827

EP - 832

AB - 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.

LA - eng

UR - http://eudml.org/doc/196017

ER -

## References

top- BEER, G., Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. Zbl0792.54008MR1269778
- COSTANTINI, C., Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc., 123 (1995), 2569-2574. Zbl0831.54014MR1273484
- COSTANTINI, C., On the hyperspace of a non-separable metric space, Proc. Amer. Math. Soc., 126 (1998), 3393-3396. Zbl0898.54012MR1618729
- VAN DOUWEN, E., The integers in topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.) North Holland, Amsterdam1984, 116-167. Zbl0561.54004MR776622
- ENGELKING, R.- MRÓWKA, On $E$-compact spaces, Bull. Acad. Pol. Sci., 6 (1958), pp. 429-439. Zbl0083.17402MR97042
- KECHRIS, A. S., Classical Descriptive Set Theory, Springer Verlag, New York, 1994. Zbl0819.04002MR1321597
- ZSILINSZKY, L., Polishness of the Wijsman topology revisited, Proc. Amer. Math. Soc., 126 (1998), pp. 3763-3765. Zbl0899.54009MR1458275

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