On Baireness of the Wijsman Topology
Abstract
We give negative answers to Questions 4.4 and 4.5 in Cao's survey on the Wijsman topology. We construct a metrizable space Y and a compatible metric d such that every finite power Yn is Baire, while (CL(Y),τW(d)) is not Baire. The construction realizes the Vietoris hyperspace of a suitable metric space as an open subspace of a ball hyperspace. We also study the set Sd of points having an open σ-d-totally bounded neighborhood. We prove that Baireness of (CL(X),τW(d)) implies Baireness of Sd, and that, whenever Sd is dense in X, the following three conditions are equivalent: (CL(X),τW(d)) is Baire, X is Baire, and Sd is Baire. As consequences, we obtain complete characterizations for locally σ-d-totally bounded metric spaces, countable metric spaces, and σ-compact metrizable spaces. In our counterexample, Se coincides with a non-dense discrete component for every compatible metric e, showing that the density assumption is essential.
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BibTeX
@misc{ryke2026baireness,
title = {{On Baireness of the Wijsman Topology}},
author = {Basile Beyer de Ryke},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35921},
eprint = {2610.35921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35921},
}
RIS
TY - UNPB
TI - On Baireness of the Wijsman Topology
AU - Basile Beyer de Ryke
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.35921
UR - https://arxiv.org/abs/2610.35921
AB - We give negative answers to Questions 4.4 and 4.5 in Cao's survey on the Wijsman topology. We construct a metrizable space Y and a compatible metric d such that every finite power Y^n is Baire, while (CL(Y),τ_W(d)) is not Baire. The construction realizes the Vietoris hyperspace of a suitable metric space as an open subspace of a ball hyperspace. We also study the set S_d of points having an open σ-d-totally bounded neighborhood. We prove that Baireness of (CL(X),τ_W(d)) implies Baireness of S_d, and that, whenever S_d is dense in X, the following three conditions are equivalent: (CL(X),τ_W(d)) is Baire, X is Baire, and S_d is Baire. As consequences, we obtain complete characterizations for locally σ-d-totally bounded metric spaces, countable metric spaces, and σ-compact metrizable spaces. In our counterexample, S_e coincides with a non-dense discrete component for every compatible metric e, showing that the density assumption is essential.
ER -