Borel and Hausdorff Hierarchies in Topological Spaces of Choquet Games and Their Effectivization

What parts of classical descriptive set theory done in Polish spaces still hold for more general topological spaces, possibly T0 or T1, but not T2 (i.e. not Hausdorff)? This question has been addressed by Victor Selivanov in a series of papers centered on algebraic domains. And recently it has been considered by Matthew de Brecht for quasi-Polish spaces, a framework that contains both countably based continuous domains and Polish spaces. In this paper we present alternative unifying topological spaces, that we call approximation spaces. They are exactly the spaces for which player Nonempty has a stationary strategy in the Choquet game. A natural proper subclass of approximation spaces coincides with the class of quasi-Polish spaces. We study the Borel and Hausdorff difference hierarchies in approximation spaces, revisiting the work done for the other topological spaces. We also consider the problem of effectivization of these results.

💡 Research Summary

The paper investigates how much of classical descriptive set theory—particularly the Borel and Hausdorff difference hierarchies—remains valid when one moves from Polish spaces to more general topological spaces that may be merely T₀ or T₁, but not necessarily Hausdorff (T₂). Building on earlier work by Victor Selivanov on algebraic domains and Matthew de Brecht on quasi‑Polish spaces, the authors introduce a new unifying class of spaces called approximation spaces. An approximation space is precisely a topological space in which the player “Nonempty” has a stationary (memory‑less) winning strategy in the Choquet game.

The authors first prove that all Polish spaces, all continuous domains, and all quasi‑Polish spaces are approximation spaces. Moreover, they identify a natural proper subclass—convergent approximation spaces—that coincides exactly with the class of quasi‑Polish spaces (Theorem 3.12). This establishes a clean game‑theoretic characterization of quasi‑Polish spaces and shows that the approximation‑space framework strictly generalizes both the domain‑theoretic and quasi‑Polish approaches.

Next, the paper develops the Borel hierarchy for arbitrary T₀ spaces. The usual definition (Σ⁰₁ = open sets, Π⁰₁ = closed sets) is retained, but Σ⁰₂ is defined as countable unions of differences of open sets (i.e., Boolean combinations of opens). This adjustment is necessary because, without the Hausdorff condition, a countable union of open sets need not be open. The authors show that Π⁰₂ consists of countable intersections of Boolean combinations of opens, and they prove that every approximation space satisfies a Π⁰₂ Baire property: any countable intersection of dense differences of opens is dense (Theorem 3.14). This generalizes the classical Baire category theorem from Polish spaces to the much broader class of approximation spaces.

The Hausdorff difference hierarchy is then introduced via the transfinite operation Dα, which iteratively forms differences of sets along an α‑sequence. The authors verify the basic closure properties of Dα and prove that for any countable α, Dα(Σ⁰β) ⊆ Δ⁰_{β+1} (Proposition 2.9). Crucially, they extend Hausdorff’s theorem: in any space with a countable basis in which every closed subspace is an approximation space, the class Δ⁰₂ coincides with the first level of the Hausdorff difference hierarchy (Theorem 3.16). This result, previously known for ω‑algebraic domains and ω‑continuous domains, now holds for all approximation spaces, and consequently for all quasi‑Polish spaces.

Section 4 revisits Selivanov’s work on domains that are not Polish, showing that the ω‑algebraicity or ω‑continuity assumptions can be replaced by mere continuity when characterizing the Hausdorff hierarchy via alternating trees. This demonstrates that the underlying combinatorial arguments are robust across a wide spectrum of spaces.

Finally, the authors turn to effectivization. They define effective approximation spaces by requiring that the underlying topology be given by a computable basis and that the stationary strategy be computable. Using the machinery of effective Borel codes and effective Hausdorff difference codes, they prove a weak effective version of Hausdorff’s theorem (Theorem 5.7): for any effective approximation space, the effective class Dα(Σ⁰₁) is contained in the effective Δ⁰₂. While a full effective Hausdorff–Kuratowski theorem remains open, this result shows that the game‑theoretic approach adapts well to computability considerations.

In summary, the paper provides a comprehensive unifying framework—approximation spaces—through which the Borel and Hausdorff hierarchies, as well as their effective counterparts, can be studied beyond Polish spaces. It bridges domain theory, quasi‑Polish topology, and descriptive set theory, and opens several avenues for future work, including a full effective Hausdorff theorem, deeper analysis of the Π⁰₂ Baire property in non‑Hausdorff settings, and applications to semantics of computation.