On Wilker-Type Properties
Abstract
Wilker-type properties record how a compact set can be split along a binary open cover, and they are precisely the continuity conditions for the union operation on open lattices, governing the topology of power and function spaces. We give a unified treatment of the four properties W, Ww, Wi and Win. We prove that Wi and Win are hereditary for arbitrary subspaces, whereas W and Ww are hereditary only for closed subspaces; all four are preserved by continuous retracts, and Ww, Wi, Win are finitely additive over topological sums. For KC-spaces Ww and Wi coincide, and spaces whose compact subspaces are Hausdorff belong to Win. Using Kovár's compact KC-space M0 and its variant QQ+, we obtain countably infinite compact KC-spaces in Wi∖W, showing Wi is strictly weaker than W. Conversely, we construct compact T1 spaces whose product fails Ww, proving the classes Ww and Wi are not reflective. We further show Vietoris constructions need not preserve Wi, and prove that Ok(M0)≅ Ck(M0,Sier) is not sober and fails property~S. Open problems are collected at the end.
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BibTeX
@misc{wang2026wilkertype,
title = {{On Wilker-Type Properties}},
author = {Wu Wang and Jiajun Tang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06060},
eprint = {2611.06060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06060},
note = {CC0},
}
RIS
TY - UNPB
TI - On Wilker-Type Properties
AU - Wu Wang
AU - Jiajun Tang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06060
UR - https://arxiv.org/abs/2611.06060
AB - Wilker-type properties record how a compact set can be split along a binary open cover, and they are precisely the continuity conditions for the union operation on open lattices, governing the topology of power and function spaces. We give a unified treatment of the four properties W, Ww, Wi and Win. We prove that Wi and Win are hereditary for arbitrary subspaces, whereas W and Ww are hereditary only for closed subspaces; all four are preserved by continuous retracts, and Ww, Wi, Win are finitely additive over topological sums. For KC-spaces Ww and Wi coincide, and spaces whose compact subspaces are Hausdorff belong to Win. Using Kovár's compact KC-space M_0 and its variant QQ^+, we obtain countably infinite compact KC-spaces in Wi∖W, showing Wi is strictly weaker than W. Conversely, we construct compact T_1 spaces whose product fails Ww, proving the classes Ww and Wi are not reflective. We further show Vietoris constructions need not preserve Wi, and prove that Ok(M_0)≅ Ck(M_0,Sier) is not sober and fails property~S. Open problems are collected at the end.
ER -