A co-sober Alexandrov space whose Scott power space is not co-sober
Abstract
We show that the Scott power construction does not preserve co-sobriety. For a dcpo P, let Fin(P) be the poset of nonempty finitely generated upper sets ordered by reverse inclusion. We prove that co-sobriety of ΣFin(P) implies co-sobriety of ΣP. The proof adapts the point-saturation and union argument used for full compact saturated power posets; the finite step follows from a result of Heckmann and Keimel. Applying this transfer to a dcpo with a non-co-sober Scott space yields a co-sober Alexandrov space whose Scott power space is not co-sober. We also construct a countable example for which the Smyth power poset is countable. In the countable example, the same poset is co-sober under the upper Vietoris topology but not under the Scott topology.
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BibTeX
@misc{ji2026cosober,
title = {{A co-sober Alexandrov space whose Scott power space is not co-sober}},
author = {Wei Ji and Xiaoquan Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01516},
eprint = {2611.01516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01516},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A co-sober Alexandrov space whose Scott power space is not co-sober
AU - Wei Ji
AU - Xiaoquan Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01516
UR - https://arxiv.org/abs/2611.01516
AB - We show that the Scott power construction does not preserve co-sobriety. For a dcpo P, let Fin(P) be the poset of nonempty finitely generated upper sets ordered by reverse inclusion. We prove that co-sobriety of ΣFin(P) implies co-sobriety of ΣP. The proof adapts the point-saturation and union argument used for full compact saturated power posets; the finite step follows from a result of Heckmann and Keimel. Applying this transfer to a dcpo with a non-co-sober Scott space yields a co-sober Alexandrov space whose Scott power space is not co-sober. We also construct a countable example for which the Smyth power poset is countable. In the countable example, the same poset is co-sober under the upper Vietoris topology but not under the Scott topology.
ER -