A model of a space X is a dcpo Y whose subspace of maximal elements is isomorphic to X. Of particular importance are those spaces that have ω-continuous models. This topic is covered in Section 7.7.2 of the book. Martin , and Mummert and Stephan  came very close to characterize them exactly. See the full post.
Polish spaces are an important class of spaces. I am dealing with them in Section 7.7 of the book. But they are Hausdorff spaces, and the purpose of the book was to explore non-Hausdorff spaces. Matthew de Brecht found a non-Hausdorff generalization of Polish spaces, which he aptly called quasi-Polish spaces. See the full post.
This is all about “Non-Hausdorff Topology and Domain Theory – Selected Topics in Point-Set Topology“, a forthcoming book, to be published by Cambridge University Press.
See the table of contents.