Wilker property, function spaces and hyperspaces
Abstract
The Wilker property requires that a compact set covered by two open sets admit compact covering pieces inside those open sets. We study this property through products, compact-open function spaces and hyperspaces. We construct countable compact Fréchet--Urysohn T1 hereditarily Wilker spaces with non-Wilker products and squares. For spaces whose compact subsets are closed, Wilker-ness is equivalent to every compact subspace being Hausdorff; this yields a positive product theorem when one Wilker factor has closed compact subsets. For an irreducible subset system H, we characterize preservation of H-sobriety by compact-open function spaces in terms of H-supercompactness of compact subsets of the domain, and show that Wilker spaces satisfy this criterion. The Hoare hyperspace of a space is Wilker exactly when binary union on the Scott space of its open-set lattice is jointly continuous for the ordinary product topology. Consequently, the Hoare construction preserves and reflects Wilker-ness on consonant bases, although preservation fails in general. We also identify core-compactness of the base with local compactness of its Hoare hyperspace and prove that the Smyth construction reflects Wilker-ness.
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BibTeX
@misc{xu2026wilker,
title = {{Wilker property, function spaces and hyperspaces}},
author = {Xiaoquan Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01547},
eprint = {2611.01547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01547},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wilker property, function spaces and hyperspaces
AU - Xiaoquan Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01547
UR - https://arxiv.org/abs/2611.01547
AB - The Wilker property requires that a compact set covered by two open sets admit compact covering pieces inside those open sets. We study this property through products, compact-open function spaces and hyperspaces. We construct countable compact Fréchet--Urysohn T_1 hereditarily Wilker spaces with non-Wilker products and squares. For spaces whose compact subsets are closed, Wilker-ness is equivalent to every compact subspace being Hausdorff; this yields a positive product theorem when one Wilker factor has closed compact subsets. For an irreducible subset system H, we characterize preservation of H-sobriety by compact-open function spaces in terms of H-supercompactness of compact subsets of the domain, and show that Wilker spaces satisfy this criterion. The Hoare hyperspace of a space is Wilker exactly when binary union on the Scott space of its open-set lattice is jointly continuous for the ordinary product topology. Consequently, the Hoare construction preserves and reflects Wilker-ness on consonant bases, although preservation fails in general. We also identify core-compactness of the base with local compactness of its Hoare hyperspace and prove that the Smyth construction reflects Wilker-ness.
ER -