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arXiv · 2608.00408

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

Xiaoyong Xi·Chong Shen·Dongsheng Zhao
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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.

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Xiaoyong Xi, Chong Shen, Dongsheng Zhao. 2026-08-01. The complete Boolean algebra of regular open sets in real line is not sober. https://arxiv.org/abs/2608.00408

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