Approximation Spaces in Descriptive Set Theory of Non-Hausdorff Topological Spaces
Approximation spaces are T0 topological spaces carrying an approximation relation on a basis, whose inclusion-decreasing chains always have non-empty intersection; equivalently, spaces in which player Nonempty has a stationary winning strategy in the Choquet game. They unify Polish spaces and continuous domains, and the convergent second-countable ones are exactly the quasi-Polish spaces. Students learn how the Borel and Hausdorff difference hierarchies behave outside metric spaces, including a Pi-0-2 Baire property and Hausdorff's characterization of Delta-0-2, and how these results effectivize.
Approximation Spaces and Borel-Hausdorff Hierarchies via Choquet Games
This paper by Veronica Becher and Serge Grigorieff investigates the extension of classical descriptive set theory (Borel and Hausdorff hierarchies) from Polish spaces to more general topological spac…