Wadge theory provides an exhaustive analysis of the topological complexity of the subsets of a zero-dimensional Polish space. Fons van Engelen pioneered its applications to topology by obtaining a classification of the zero-dimensional homogeneous Borel spaces (recall that a space $X$ is homogeneous if for all $x,y\in X$ there exists a homeomorphism $h:X\longrightarrow X$ such that $h(x)=y$).

As a corollary, he showed that all such spaces (apart from trivial

exceptions) are in fact strongly homogeneous (recall that a space $X$ is strongly homogeneous if all non-empty clopen subspaces of $X$ are homeomorphic to each other).

In a joint work with the other members of the “Wadge Brigade” (namely, Raphaël Carroy and Sandra Müller), we showed that this last result extends beyond the Borel realm if one assumes AD. We intend to sketch the proof of this theorem, with a view towards a complete classification of the zero-dimensional homogeneous spaces under AD.