In my first post on ideal domains, I thought I would be able to extend Keye Martin’s result from metric to quasi-metric spaces. That was more complicated than what I had thought.

Along my journey, I (re)discovered a few results, some old, some new, on ideal completion remainders—namely, the spaces you get by taking the ideal completion of a poset *P*, and substracting *P* off—and on the related notion of sobrification remainders. That may seem like silly notions to you, and I certainly thought so until recently. But they seem to crop up from time to time.

I will show you that *every* T_{0} space occurs as a sobrification remainder (a result due to R.-E. Hoffmann), and I will give you the rough idea of a proof that the ideal completion remainders of countable posets are exactly the quasi-Polish spaces (a result due to M. de Brecht). I will also describe an intriguing result on wqos and bqos due to Y. Péquignot and R. Carroy. Read the full post.