Quasi-polish spaces

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 T3 spaces, they are exactly those which have ω-continuous 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 “non-Hausdorff topology”, right?

Well, first, what I mean with non-Hausdorff 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 non-Hausdorff 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 quasi-Polish 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 quasi-Polish spaces as those which underly a Smyth-complete separable quasi-metric space. The move to quasi-metric spaces was expected, but there were (at least) two distinct, competing notions of completeness that one may want to use, Smyth-completeness and Yoneda-completeness (see Chapter 7 in the book). It seems that Smyth-completeness 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 Gδ.  (A Gδ subset is the intersection of countably many open subsets.)  A variant is that the Polish spaces are exactly the Gδ subsets of the so-called Hilbert cube, up to homeomorphism.  Matthew shows that the same holds for quasi-Polish spaces, provided we replace the Gδ subsets by the Π02 subsets, and the Hilbert cube by the powerset of the natural numbers, with a suitable quasi-metric.  These are Corollaries 23 and 24 in Matthew’s paper.  (In this non-Hausdorff setting, the correct definition of Π02 is slightly more complex than usual.  As usual, the Π02 subsets are the complements of Σ02 subsets.  But the Σ02 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 Choquet-complete.  The latter is a game-theoretic 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 non-empty, and a space is Choquet-complete if α has a winning strategy.  There are several variants of this notion.  One is convergence Choquet-completeness, 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 countably-based T0 space, being quasi-Polish is equivalently to being convergence Choquet-complete.
    • 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 Gδ superset; here again Gδ subsets must be replaced by Π02 subsets), Martin’s Theorem (already mentioned above), and much more.

Jean Goubault-Larrecqjgl-2011

[1] Matthew de Brecht.  Quasi-Polish Spaces.  arXiv:1108.1445, Aug 6, 2011.