# Wijsman hyperspaces of non-separable metric spaces

Fundamenta Mathematicae (2015)

• Volume: 228, Issue: 1, page 63-79
• ISSN: 0016-2736

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## Abstract

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Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology ${\tau }_{W\left(\rho \right)}$. It is known that $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ 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 $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL\left(X\right),{\tau }_{W\left(\rho \right)}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

## How to cite

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Rodrigo Hernández-Gutiérrez, and Paul J. Szeptycki. "Wijsman hyperspaces of non-separable metric spaces." Fundamenta Mathematicae 228.1 (2015): 63-79. <http://eudml.org/doc/283021>.

@article{RodrigoHernández2015,
abstract = {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 $⟨CL(X),τ_\{W(ρ)\}⟩$ 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 $⟨CL(X),τ_\{W(ρ)\}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL(X),τ_\{W(ρ)\}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.},
author = {Rodrigo Hernández-Gutiérrez, Paul J. Szeptycki},
journal = {Fundamenta Mathematicae},
keywords = {metric space; hyperspace; Wijsman; normal; separable},
language = {eng},
number = {1},
pages = {63-79},
title = {Wijsman hyperspaces of non-separable metric spaces},
url = {http://eudml.org/doc/283021},
volume = {228},
year = {2015},
}

TY - JOUR
AU - Rodrigo Hernández-Gutiérrez
AU - Paul J. Szeptycki
TI - Wijsman hyperspaces of non-separable metric spaces
JO - Fundamenta Mathematicae
PY - 2015
VL - 228
IS - 1
SP - 63
EP - 79
AB - 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 $⟨CL(X),τ_{W(ρ)}⟩$ 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 $⟨CL(X),τ_{W(ρ)}⟩$ being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then $⟨CL(X),τ_{W(ρ)}⟩$ is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.
LA - eng
KW - metric space; hyperspace; Wijsman; normal; separable
UR - http://eudml.org/doc/283021
ER -

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