Remainders, bqos, and quasi-Polish spaces again

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 T0 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.

Ideal models II

Last time, we have seen that every completely metrizable space X has an ideal model, that is, that X can be embedded into an ideal domain Y in such a way that we can equate X with the subspace of maximal elements of Y.

We have also seen the converse to that: if X is a metrizable space with an ideal model, then X is completely metrizable.

But we had skipped an essential ingredient: that the set X of maximal elements of an ideal model Y is a Gδ subset of Y.  This is true, but complicated.  As I have already said last time, we shall do something slightly simpler.  See the full post.