# Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

### (with R. Carroy and A. Medini)

Accepted at the Journal of Mathematical Logic. 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.