Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

Journal of Mathematical Logic 20 (3):2050015 (2020)
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Abstract

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

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Sandra Eleonore Müller
Ludwig Maximilians Universität, München

Citations of this work

Constructing wadge classes.Raphaël Carroy, Andrea Medini & Sandra Müller - 2022 - Bulletin of Symbolic Logic 28 (2):207-257.
Zero-dimensional σ-homogeneous spaces.Andrea Medini & Zoltán Vidnyánszky - 2024 - Annals of Pure and Applied Logic 175 (1):103331.

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References found in this work

Infinite combinatorics and definability.Arnold W. Miller - 1989 - Annals of Pure and Applied Logic 41 (2):179-203.
On Borel ideals.Fons van Engelen - 1994 - Annals of Pure and Applied Logic 70 (2):177-203.
Borel-amenable reducibilities for sets of reals.Luca Motto Ros - 2009 - Journal of Symbolic Logic 74 (1):27-49.
Between Polish and completely Baire.Andrea Medini & Lyubomyr Zdomskyy - 2015 - Archive for Mathematical Logic 54 (1-2):231-245.
Wadge hierarchy of differences of co-analytic sets.Kevin Fournier - 2016 - Journal of Symbolic Logic 81 (1):201-215.

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