A countable metric space whose Hoare power space is not co-sober
Abstract
Xu asked whether the Hoare power space of a co-sober space, and in particular of a T2-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable T1 space X whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space M onto X. Direct and inverse images then make the Scott space Σ𝒪(X) a retract of Σ𝒪(M), so Σ𝒪(M) is not sober. Finally, we show that, for every T0-space Y, the Hoare power space PH(Y) is co-sober if and only if the Scott space Σ𝒪(Y) is sober. Consequently, PH(M) is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.
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BibTeX
@misc{xu2026countable,
title = {{A countable metric space whose Hoare power space is not co-sober}},
author = {Xiaoquan Xu and Wei Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05868},
eprint = {2610.05868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05868},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A countable metric space whose Hoare power space is not co-sober
AU - Xiaoquan Xu
AU - Wei Ji
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05868
UR - https://arxiv.org/abs/2610.05868
AB - Xu asked whether the Hoare power space of a co-sober space, and in particular of a T_2-space or a metric space, must be co-sober. We give a negative answer. He and Zhao recently constructed a countable T_1 space X whose lattice of open sets has a non-sober Scott topology. We observe that their topology has a countable subbase. The classical Ponomarev construction then yields an open continuous surjection from a countable zero-dimensional metrizable space M onto X. Direct and inverse images then make the Scott space Σ𝒪(X) a retract of Σ𝒪(M), so Σ𝒪(M) is not sober. Finally, we show that, for every T_0-space Y, the Hoare power space PH(Y) is co-sober if and only if the Scott space Σ𝒪(Y) is sober. Consequently, PH(M) is not co-sober. Therefore, the same example answers negatively a question of He and Zhao about Scott sobriety of the lattices of open sets of countable Hausdorff spaces.
ER -