• Corpus ID: 248496504

Not every countable complete lattice is sober

@inproceedings{Miao2022NotEC,
  title={Not every countable complete lattice is sober},
  author={Hualin Miao and Xiaoyong Xi and Qingguo Li and Dongsheng Zhao},
  year={2022}
}

• Hualin Miao, Xiaoyong Xi, +1 author Dongsheng Zhao
• Published 30 April 2022
• Mathematics

The study of the sobriety of Scott spaces has got an relative long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Not long after, Isbell gave a complete lattice with non-sober Scott space. Based on Isbell’s example, Xu, Xi and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. Let Σ P be the Scott space of poset P . In this paper, we first prove that the topology of the product space Σ P × Σ Q coincides with the Scott topology on the product poset P × Q if the set Id ( P ) and Id ( Q ) of all non-trivial ideals of posets P and Q are both countable. Based on this result, we deduce that a directed complete poset P has a sober Scott space, if Id ( P ) is countable and the space Σ P is coherent and well-filtered. Thus a complete lattice L with Id ( L ) countable has a sober Scott space. Making use the obtained results, we then construct a countable complete lattice whose Scott space is non-sober and thus give a negative answer to Jung’s problem. 2000 MSC: 54B20,