Note on the Wijsman hyperspaces of completely metrizable spaces

J. Chaber; R. Pol

Bollettino dell'Unione Matematica Italiana (2002)

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

Abstract

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We consider the hyperspace C ⁢ L ⁢ X of nonempty closed subsets of completely metrizable space X endowed with the Wijsman topologies τ W d . If X is separable and d , e are two metrics generating the topology of X , every countable set closed in C ⁢ L ⁢ X , τ W e has isolated points in C ⁢ L ⁢ X , τ W d . For d = e , this implies a theorem of Costantini on topological completeness of C ⁢ L ⁢ X , τ W d . We show that for nonseparable X the hyperspace C ⁢ L ⁢ X , τ W d may contain a closed copy of the rationals. This answers a question of Zsilinszky.

How to cite

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Chaber, 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

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  1. BEER, G., Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. Zbl0792.54008MR1269778
  2. COSTANTINI, C., Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc., 123 (1995), 2569-2574. Zbl0831.54014MR1273484
  3. COSTANTINI, C., On the hyperspace of a non-separable metric space, Proc. Amer. Math. Soc., 126 (1998), 3393-3396. Zbl0898.54012MR1618729
  4. 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
  5. ENGELKING, R.- MRÓWKA, On E -compact spaces, Bull. Acad. Pol. Sci., 6 (1958), pp. 429-439. Zbl0083.17402MR97042
  6. KECHRIS, A. S., Classical Descriptive Set Theory, Springer Verlag, New York, 1994. Zbl0819.04002MR1321597
  7. ZSILINSZKY, L., Polishness of the Wijsman topology revisited, Proc. Amer. Math. Soc., 126 (1998), pp. 3763-3765. Zbl0899.54009MR1458275

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