The complete Boolean algebra of regular open sets in real line is not sober

Xiaoyong Xi; Chong Shen; Dongsheng Zhao

The complete Boolean algebra of regular open sets in real line is not sober

General Topology 2026-08-01 v1

Authors: Xiaoyong Xi , Chong Shen , Dongsheng Zhao

Abstract

Whether the Scott space of the complete Boolean algebra $\RO(\bbR)$ of regular open subsets in the real line is sober was explicitly recorded as an open problem by Xu and Zhao and was reiterated in a later survey. We give a negative answer to this problem in this paper.

Keywords

Scott space Boolean algebra regular open sets sober

Cite

@article{arxiv.2608.00408,
  title  = {The complete Boolean algebra of regular open sets in real line is not sober},
  author = {Xiaoyong Xi and Chong Shen and Dongsheng Zhao},
  journal= {arXiv preprint arXiv:2608.00408},
  year   = {2026}
}