A complete classification of the zero-dimensional homogeneous spaces under determinacy
Andrea Medini
TL;DR
The paper delivers a complete classification of zero-dimensional homogeneous spaces under determinacy, extending van Engelen’s Borel results beyond the Borel realm. It develops a robust Wadge-theoretic framework, introducing level and type for good Wadge classes and proving key closure and expansion properties that enable existence and uniqueness results for spaces of exact complexity. A central outcome is that, under suitable determinacy assumptions, every high-complexity zero-dimensional homogeneous space is homeomorphic to a semifilter on $oldsymbol{ olinebreak[4]}oldsymbol{ ext{ω}}$, and filters on $oldsymbol{ ext{ω}}$ are likewise classifiable by exact complexity with purely topological characterizations. The work also yields independent Wadge-theoretic results of broader interest and clarifies the interplay between homogeneity, Baire category, and topological group structures in this setting, thereby providing a comprehensive atlas of zero-dimensional homogeneous spaces under determinacy with precise existence/uniqueness statements. The findings have potential implications for the study of topological groups and for descriptive set-theoretic classifications beyond the Borel level, under AD-compatible axioms.
Abstract
All spaces are assumed to be separable and metrizable. We give a complete classification of the zero-dimensional homogeneous spaces, under the Axiom of Determinacy. This classification is expressed in terms of topological complexity (in the sense of Wadge theory) and Baire category. In the same spirit, we also give a complete classification of the filters on $ω$ up to homeomorphism. As byproducts, we obtain purely topological characterizations of the semifilters and filters on $ω$. The Borel versions of these results are in almost all cases due to Fons van Engelen. Along the way, we obtain Wadge-theoretic results of independent interest, especially regarding closure properties.