Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-Gutiérrez; Paul J. Szeptycki

Rodrigo Hernández-Gutiérrez ; Paul J. Szeptycki

Fundamenta Mathematicae, Tome 228 (2015), p. 63-79 / Harvested from The Polish Digital Mathematics Library

Résumé

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology $τ_{W (ρ)}$ . It is known that $⟨ C L (X), τ_{W (ρ)} ⟩$ is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to $⟨ C L (X), τ_{W (ρ)} ⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨ C L (X), τ_{W (ρ)} ⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

Publié le : 2015-01-01

Zbl 1316.54004

EUDML-ID : urn:eudml:doc:283021

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     author = {Rodrigo Hern\'andez-Guti\'errez and Paul J. Szeptycki},
     title = {Wijsman hyperspaces of non-separable metric spaces},
     journal = {Fundamenta Mathematicae},
     volume = {228},
     year = {2015},
     pages = {63-79},
     zbl = {1316.54004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-1-5}
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Rodrigo Hernández-Gutiérrez; Paul J. Szeptycki. Wijsman hyperspaces of non-separable metric spaces. Fundamenta Mathematicae, Tome 228 (2015) pp. 63-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm228-1-5/