Zhao, Xi and Chen’s well-filtered, non-sober dcpo

There are several known examples of dcpos that are well-filtered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This is a very simple modification of Johnstone’s non-sober dcpo J. Contrarily to Xiaodong Jia’s dcpo (and to J), it is uncountable, but it may be easier to see why it must be well-filtered: everything mostly boils down to a cardinality argument, or rather, as I will argue, to the properties of so-called regular ordinals. Read the full post.