Isbell’s non sober complete lattice

Johnstone space J (1981) is the most famous example of a non-sober dcpo. In 1982, Isbell came up with a non-sober complete lattice. His construction is so complex that most authors use it as a black box. I would like to explain how Isbell’s non-sober complete lattice is constructed. As you can expect, this is a pretty clever construction, but I claim this is understandable. I will conclude with remarkable results of Xu, Xi, and Zhao (2019), who proved that there even exists a non-sober frame. Read the full post.