Ideal domains I

A few months ago, Keye Martin drew my attention to his results on so-called ideal models of spaces [1].  Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal.  Despite this, Keye Martin was able to show that: (1) every space that has an ω-continuous model has an ideal model, that is, a model that is an ideal domain; (2) the metrizable spaces that have an ideal model are exactly the completely metrizable spaces.

I will try to expose a few of his ideas here.  I will probably betray him a lot.  For example, I will not talk about measurements (one of Keye’s inventions), and I will not stress the role of Choquet-completeness to go beyond “Lawson at the top” domains, or the role of Gδ subsets so much.

Last minute update: I had also tried to extend whatever I could to the case of quasi-metric, not just metric, spaces, but I did not manage to do so. Read the (corrected) full post.