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.