Sequential convergence in the lower topology and co-sobriety of Scott spaces
Abstract
We investigate how order structure controls sequential convergence in the lower topology of a poset and how this, in turn, governs topological duality and co-sobriety of Scott spaces. We first give a purely order-theoretic characterization of lower convergence. For complete lattices this yields a lower-limsup formula, and on complete chains the lower limsup coincides with the usual lattice limsup. We also characterize first countability of the lower topology at a point by the existence of a countable coinitial subset outside its principal ideal. We prove that every compact saturated subset of a Scott space is sequentially closed in the lower topology and deduce that, for every poset, the lower topology and the co-compact (de Groot dual) topology of its Scott space have exactly the same convergent sequences and limits. For coherent Scott spaces we identify the lower closure of a k-irreducible compact saturated set with its double-cone closure and obtain an infimum criterion for co-sobriety. Consequently, every bounded-complete dcpo with sequential lower topology has a co-sober Scott space. In particular, every countable bounded-complete dcpo, and hence every countable complete lattice, has a co-sober Scott space.
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BibTeX
@misc{xu2026sequential,
title = {{Sequential convergence in the lower topology and co-sobriety of Scott spaces}},
author = {Xiaoquan Xu and Wei Ji},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01487},
eprint = {2611.01487},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01487},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sequential convergence in the lower topology and co-sobriety of Scott spaces
AU - Xiaoquan Xu
AU - Wei Ji
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01487
UR - https://arxiv.org/abs/2611.01487
AB - We investigate how order structure controls sequential convergence in the lower topology of a poset and how this, in turn, governs topological duality and co-sobriety of Scott spaces. We first give a purely order-theoretic characterization of lower convergence. For complete lattices this yields a lower-limsup formula, and on complete chains the lower limsup coincides with the usual lattice limsup. We also characterize first countability of the lower topology at a point by the existence of a countable coinitial subset outside its principal ideal. We prove that every compact saturated subset of a Scott space is sequentially closed in the lower topology and deduce that, for every poset, the lower topology and the co-compact (de Groot dual) topology of its Scott space have exactly the same convergent sequences and limits. For coherent Scott spaces we identify the lower closure of a k-irreducible compact saturated set with its double-cone closure and obtain an infimum criterion for co-sobriety. Consequently, every bounded-complete dcpo with sequential lower topology has a co-sober Scott space. In particular, every countable bounded-complete dcpo, and hence every countable complete lattice, has a co-sober Scott space.
ER -