# 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} }

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…

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