All spaces are assumed to be separable and metrizable. A space \(X\) is homogeneous if for all \(x,y\in X\) there exists a homeomorphism \(h:X\longrightarrow X\) such that \(h(x)=y\). A space \(X\) is strongly homogeneous if all non-empty clopen subspaces of \(X\) are homeomorphic to each other. We will show that, under the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (with the trivial exception of locally compact spaces). This extends results of van Engelen and complements a result of van Douwen. Our main tool will be Wadge theory, which provides an exhaustive analysis of the topological complexity of the subsets of \(2\omega\).
This is joint work with Raphaël Carroy and Sandra Müller.