In his PhD thesis Wadge characterized the notion of continuous reducibility on the Baire space ${}^\omega\omega$ in form of a game and analyzed it in a systematic way. He defined a refinement of the Borel hierarchy, called the Wadge hierarchy, showed that it is well-founded, and (assuming determinacy for Borel sets) proved that every Borel pointclass appears in this classification. Later Louveau found a description of all levels in the Borel Wadge hierarchy using Boolean operations on sets. Fons van Engelen used this description to analyze Borel homogeneous spaces.

We use Wadge theory to show that under suitable determinacy assumptions, every definable zero-dimensional space is h-homogeneous. This extends a result of van Engelen, who obtained the corresponding result (in ZFC) for Borel spaces.

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