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arXiv · 2408.08587

Products of two sober dcpo's need not be sober

Hualin Miao·Xiaoyong Xi·Xiaodong Jia·Qingguo Li·Dongsheng Zhao
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Abstract

We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober.

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Hualin Miao, Xiaoyong Xi, Xiaodong Jia, Qingguo Li, Dongsheng Zhao. 2025-12-22. Products of two sober dcpo's need not be sober. https://arxiv.org/abs/2408.08587

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