Note on the Wijsman hyperspaces of completely metrizable spaces

Chaber, J.; Pol, R.

Chaber, J. ; Pol, R.

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 827-832 / Harvested from Biblioteca Digitale Italiana di Matematica

Access to full text
Full (PDF)
Full (DjVu)

Abstract
Résumé

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.

Consideriamo sugli spazi $C L (X)$ dei sottoinsiemi chiusi e non vuoti di uno spazio $X$ completamente metrizzabile la topologia di Wijsman $τ_{W_{d}}$ . Se $X$ è separabile, mostriamo che, per ogni metrica $d$ , $e$ su $X$ , ogni insieme chiuso e numerabile in $(C L (X), τ_{W_{e}})$ ha punti isolati in $(C L (X), τ_{W_{d}})$ . Se $d = e$ , questo implica il teorema di Costantini sulla completezza topologica di $(C L (X), τ_{W_{d}})$ . Per $X$ non-separabili, rispondiamo ad una questione sollevata da Zsilinszky, mostrando che in molti casi gli spazi $(C L (X), τ_{W_{d}})$ contengono copie chiuse dei razionali.

Publié le : 2002-10-01

MR 1934383 | Zbl 1098.54006

@article{BUMI_2002_8_5B_3_827_0,
     author = {J. Chaber and R. Pol},
     title = {Note on the Wijsman hyperspaces of completely metrizable spaces},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {827-832},
     zbl = {1098.54006},
     mrnumber = {1934383},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_3_827_0}
}

Chaber, J.; Pol, R. Note on the Wijsman hyperspaces of completely metrizable spaces. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 827-832. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_3_827_0/

Bibliographie

[1] Beer, G., Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. | MR 1269778 | Zbl 0792.54008

[2] Costantini, C., Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc., 123 (1995), 2569-2574. | MR 1273484 | Zbl 0831.54014

[3] Costantini, C., On the hyperspace of a non-separable metric space, Proc. Amer. Math. Soc., 126 (1998), 3393-3396. | MR 1618729 | Zbl 0898.54012

[4] Van Douwen, E., The integers in topology, Handbook of Set-Theoretic Topology (K. Kunen and J. E. Vaughan, eds.) North Holland, Amsterdam 1984, 116-167. | MR 776622 | Zbl 0561.54004

[5] Engelking, R.-Mrówka, On $E$ -compact spaces, Bull. Acad. Pol. Sci., 6 (1958), pp. 429-439. | MR 97042 | Zbl 0083.17402

[6] Kechris, A. S., Classical Descriptive Set Theory, Springer Verlag, New York, 1994. | MR 1321597 | Zbl 0819.04002

[7] Zsilinszky, L., Polishness of the Wijsman topology revisited, Proc. Amer. Math. Soc., 126 (1998), pp. 3763-3765. | MR 1458275 | Zbl 0899.54009