X. Jia’s well-filtered, non-sober dcpo

[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not well-filtered either. Is there a dcpo that is not sober but is well-filtered? That is true, and the first one who found an example is Hui Kou. Since then, Xi and Zhao have also given another example, and I would like to describe another example of such a dcpo, due to Xiaodong Jia in his PhD thesis. Both Xi and Zhao’s example and Jia’s example are pretty simple spaces, but X. Jia’s example is countable. Read the full post.