Kernel retracts of co-sober spaces
Abstract
Kernel retracts of co-sober spaces need not be co-sober, even when the ambient space is countably infinite and compact. We construct a co-sober T0-space Y and a continuous idempotent map k: Y→ Y satisfying k(y)≤Y y in the specialization order, whose image with the subspace topology is not co-sober. The construction starts from a countable compact T1 KC anti-Hausdorff space of van Douwen type and adjoins an infinite chain. We classify the k-irreducible compact saturated subsets of the extension and show that each is a point saturation. The kernel operator collapses the added chain to a distinguished point while fixing the original space, thereby recovering the non-co-sober seed as its image. This answers negatively two questions of Xu and Zhao, concerning continuous kernel operators and arbitrary retracts, and contrasts with the known preservation of co-sobriety by continuous closure operators.
Related
- Symmetric products and finite subset spaces of absolute Lipschitz retracts
- Applications of sequential spaces to Smyth power spaces and sober spaces
- Products of co-sober spaces need not be co-sober
- Function spaces with compact-open topology need not be sober
- H-sober spaces and Hofmann-Mislove type theorems
BibTeX
@misc{xu2026kernel,
title = {{Kernel retracts of co-sober spaces}},
author = {Xiaoquan Xu and Fengshuo Xu and Haitao Ji and Dong Qiu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01531},
eprint = {2611.01531},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01531},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kernel retracts of co-sober spaces
AU - Xiaoquan Xu
AU - Fengshuo Xu
AU - Haitao Ji
AU - Dong Qiu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01531
UR - https://arxiv.org/abs/2611.01531
AB - Kernel retracts of co-sober spaces need not be co-sober, even when the ambient space is countably infinite and compact. We construct a co-sober T_0-space Y and a continuous idempotent map k: Y→ Y satisfying k(y)≤_Y y in the specialization order, whose image with the subspace topology is not co-sober. The construction starts from a countable compact T_1 KC anti-Hausdorff space of van Douwen type and adjoins an infinite chain. We classify the k-irreducible compact saturated subsets of the extension and show that each is a point saturation. The kernel operator collapses the added chain to a distinguished point while fixing the original space, thereby recovering the non-co-sober seed as its image. This answers negatively two questions of Xu and Zhao, concerning continuous kernel operators and arbitrary retracts, and contrasts with the known preservation of co-sobriety by continuous closure operators.
ER -