Products of co-sober spaces need not be co-sober
Abstract
We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset A of a T1-space (X,τ), we consider the topology consisting of the co-finite open sets together with the τ-open sets disjoint from A. We determine its k-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton k-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář--van Douwen space gives two countable compact Fréchet--Urysohn T1 anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet--Urysohn T1 anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of T0-spaces.
Related
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BibTeX
@misc{xu2026products,
title = {{Products of co-sober spaces need not be co-sober}},
author = {Xiaoquan Xu and Wei Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14434},
eprint = {2610.14434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14434},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Products of co-sober spaces need not be co-sober
AU - Xiaoquan Xu
AU - Wei Ji
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14434
UR - https://arxiv.org/abs/2610.14434
AB - We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset A of a T_1-space (X,τ), we consider the topology consisting of the co-finite open sets together with the τ-open sets disjoint from A. We determine its k-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton k-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář--van Douwen space gives two countable compact Fréchet--Urysohn T_1 anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet--Urysohn T_1 anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of T_0-spaces.
ER -