Polish spaces are an important class of spaces. I am dealing with them in Section 7.7.
Most notably, they are the kinds of spaces where topological measure theory works best. But they also occur naturally in several other situations. For example, among all T_{3} spaces, they are exactly those which have ωcontinuous models. This is Martin’s first theorem (Theorem 7.7.23 in the book), and I’m explaining what this is about in Section 7.7.2.
I had a conundrum with Polish spaces. These are Hausdorff spaces, and the title of the book says “nonHausdorff topology”, right?
Well, first, what I mean with nonHausdorff topology is topology that is not restricted to Hausdorff spaces only. So I never felt forced to only deal with spaces that are not Hausdorff.
Still, is there a nonHausdorff generalization of Polish spaces? I have tried to think about it for some time, but could not find anything convincing.
The news is that Matthew de Brecht managed to find one, which he called quasiPolish spaces. He even went as far as proving the analogs of all the classical theorems on Polish spaces in his new setting. (Thanks to Klaus Keimel, who pointed me to this piece of work.)
His paper can be found on arXiv [1]. Here are a few highlights:

 Recall that a Polish space is the topological space that underlies a complete separable metric space. Matthew defines quasiPolish spaces as those which underly a Smythcomplete separable quasimetric space. The move to quasimetric spaces was expected, but there were (at least) two distinct, competing notions of completeness that one may want to use, Smythcompleteness and Yonedacompleteness (see Chapter 7 in the book). It seems that Smythcompleteness is the right one in this setting.
 A theorem by Alexandroff states that it is equivalent, for a subspace of a Polish space, to be Polish or to be G_{δ}. (A G_{δ }subset is the intersection of countably many open subsets.) A variant is that the Polish spaces are exactly the G_{δ }subsets of the socalled Hilbert cube, up to homeomorphism. Matthew shows that the same holds for quasiPolish spaces, provided we replace the G_{δ }subsets by the Π^{0}_{2} subsets, and the Hilbert cube by the powerset of the natural numbers, with a suitable quasimetric. These are Corollaries 23 and 24 in Matthew’s paper. (In this nonHausdorff setting, the correct definition of Π^{0}_{2} is slightly more complex than usual. As usual, the Π^{0}_{2} subsets are the complements of Σ^{0}_{2} subsets. But the Σ^{0}_{2} subsets must now be defined as the countable unions of crescents, not opens. A crescent is the intersection of an open and a closed subset.)
 In the classical case, Choquet’s Theorem (see Theorem 7.6.15 in the book) implies that the Polish subspaces of a Polish space are exactly those that are Choquetcomplete. The latter is a gametheoretic notion of completeness, where two players α and β take turns and pick smaller and smaller open subsets, and points inside them. α wins if the intersection of all these opens is nonempty, and a space is Choquetcomplete if α has a winning strategy. There are several variants of this notion. One is convergence Choquetcompleteness, introduced by Dorais and Mummert in 2010, and independently by Matthew (see Exercises 7.6.5, 7.6.6 in the book). Matthew shows that, for a countablybased T_{0} space, being quasiPolish is equivalently to being convergence Choquetcomplete.
 Matthew also proves extensions of many other results: Lavrentiev’s Theorem (classically: any continuous map from a complete subset of a metric space to a metric space extends to a G_{δ} superset; here again G_{δ }subsets must be replaced by Π^{0}_{2} subsets), Martin’s Theorem (already mentioned above), and much more.
— Jean GoubaultLarrecq (October 5th, 2012)
[1] Matthew de Brecht. QuasiPolish Spaces. arXiv:1108.1445, Aug 6, 2011.