Quasi-Polish Spaces

We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is level \Pi_2 in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an \omega-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every \omega-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff-Kuratowski theorem generalizes to all quasi-Polish spaces.

💡 Research Summary

The paper introduces and develops a descriptive set theory for a class of non‑Hausdorff spaces called quasi‑Polish spaces. A quasi‑Polish space is defined as a countably based topological space that admits a complete quasi‑metric: a distance‑like function satisfying the triangle inequality and the condition that (x=y) iff (d(x,y)=d(y,x)=0), but without requiring symmetry. Completeness is understood in the usual Cauchy‑sequence sense, ensuring that the induced topology is completely quasi‑metrizable.

The authors first explain why the classical Borel hierarchy (where (\Sigma^0_2) are (F_\sigma) sets and (\Pi^0_2) are (G_\delta) sets) fails for many non‑metrizable spaces, especially those arising in domain theory. They adopt Selivanov’s generalized Borel hierarchy, defined recursively for all countable ordinals (\alpha<\omega_1). This hierarchy retains closure under countable unions/intersections and behaves well under continuous preimages, even in T₀ spaces.

Key results are:

1. Subspace Characterisation – A subspace of a quasi‑Polish space is quasi‑Polish exactly when it is a (\Pi^0_2) subset of the ambient space. Thus the second level of the Borel hierarchy plays the same role as closedness does in Polish spaces.

2. Computability (TTE) Characterisation – Within Type‑2 Theory of Effectivity, a countably based space has an admissible representation whose domain is a Polish space if and only if the space is quasi‑Polish. Hence quasi‑Polish spaces are precisely those that are “effectively Polish” from the viewpoint of computable analysis.

3. Domain‑Theoretic Characterisation – Every (\omega)-continuous domain (D) has a subspace consisting of its non‑compact elements; equipped with the Scott topology, this subspace is homeomorphic to a quasi‑Polish space, and conversely every quasi‑Polish space arises this way. Consequently all (\omega)-continuous (in particular (\omega)-algebraic) domains are quasi‑Polish.

4. Relation to Classical Polish Spaces – If a quasi‑Polish space is metrizable, then it is exactly a Polish space. Moreover, every countably based locally compact sober space is quasi‑Polish, providing a broad class of examples beyond metric spaces.

5. Borel Hierarchy Non‑Collapse – For any uncountable quasi‑Polish space, the generalized Borel hierarchy does not collapse: for each countable ordinal (\alpha) the classes (\Sigma^0_\alpha) and (\Pi^0_\alpha) are distinct, and every Borel set belongs to a unique minimal level.

6. Hausdorff‑Kuratowski Theorem Extension – The classical theorem stating that the Borel hierarchy on a Polish space has exactly (\omega_1) distinct levels is proved to hold for all quasi‑Polish spaces. This includes the usual relationships (\Delta^0_{\alpha+1} = \Sigma^0_{\alpha+1} \cap \Pi^0_{\alpha+1}) and the fact that each Borel set can be expressed as a countable Boolean combination of open sets at some finite stage.

The paper also discusses bicomplete quasi‑metrics (those for which both the quasi‑metric and its conjugate are complete) and shows that a countably based space admitting such a structure is exactly a (\Pi^0_3) subset of a quasi‑Polish space, answering a question posed by Selivanov.

Methodologically, the authors blend techniques from general topology, descriptive set theory, domain theory, and computable analysis. They provide numerous propositions establishing the basic closure properties of the generalized Borel classes, prove that singletons and diagonals are (\Pi^0_2) in any countably based (T_0) space, and use these facts to handle equality of continuous functions and related constructions.

In the concluding sections, the authors outline further research directions: a finer analysis of admissible representations in higher Borel levels, extensions to non‑countably based spaces, and connections with (\omega)-ideal domains as raised by K. Martin.

Overall, the work offers a unifying framework that simultaneously generalises classical Polish descriptive set theory and the domain‑theoretic approach to non‑Hausdorff spaces, establishing quasi‑Polish spaces as the natural setting for a robust, non‑metrizable descriptive set theory.