# 2018 seminar talk: Homogeneous spaces and Wadge theory

Talk held by Andrea Medini (KGRC) at the KGRC seminar on 2018-06-07.

### Abstract

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.