Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

(with R. Carroy and A. Medini)

Journal of Mathematical Logic. Volume 20, Issue 3, March 2020. 2050015.
DOI: 10.1142/S0219061320500154. PDF. arXiv. Bibtex.

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 (that is, 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) and complements a result of van Douwen.