Every countable meet-continuous lattice is Scott sober
Abstract
We prove that every countable meet-continuous lattice is Scott sober. For any family of such lattices, the Scott topology on the product equals the product of the factor Scott topologies, and the product Scott space is sober. These results answer affirmatively Questions 7.1 and 7.2 of Xu (arXiv:2609.18032v1), extending the countable-frame results beyond finite distributivity. A diagonal argument on meet coordinates extracts directed punctured intervals from nonprincipal ideals. Join translates of these intervals form a countable family detecting Scott openness and yield Scott open rectangles by successive finite choices. We also show that the cardinal spectrum of Scott non-sober meet-continuous lattices is upward closed and, under the Continuum Hypothesis, consists of all uncountable cardinals.
Related
- Scott products and sobriety of countable meet-continuous dcpos
- The product of two Scott sober complete lattices is not always Scott sober
- A co-sober Alexandrov space whose Scott power space is not co-sober
- A countable metric space whose Hoare power space is not co-sober
- Scott topologies on products of countable complete Heyting algebras
BibTeX
@misc{xu2026every,
title = {{Every countable meet-continuous lattice is Scott sober}},
author = {Xiaoquan Xu and Wei Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19612},
eprint = {2609.19612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19612},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Every countable meet-continuous lattice is Scott sober
AU - Xiaoquan Xu
AU - Wei Ji
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19612
UR - https://arxiv.org/abs/2609.19612
AB - We prove that every countable meet-continuous lattice is Scott sober. For any family of such lattices, the Scott topology on the product equals the product of the factor Scott topologies, and the product Scott space is sober. These results answer affirmatively Questions 7.1 and 7.2 of Xu (arXiv:2609.18032v1), extending the countable-frame results beyond finite distributivity. A diagonal argument on meet coordinates extracts directed punctured intervals from nonprincipal ideals. Join translates of these intervals form a countable family detecting Scott openness and yield Scott open rectangles by successive finite choices. We also show that the cardinal spectrum of Scott non-sober meet-continuous lattices is upward closed and, under the Continuum Hypothesis, consists of all uncountable cardinals.
ER -