**"Zero-dimensional $\sigma$-homogeneous spaces"**
#### Medini, AndreaAll spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is $\sigma$-homogeneous. Inspired by this theorem, we obtained the following results: (1) Assuming $\mathsf{AD}$, every zero-dimensional space is $\sigma$-homogeneous, (2) Assuming $\mathsf{AC}$, there exists a zero-dimensional space that is not $\sigma$-homogeneous, (3) Assuming $\mathsf{V=L}$, there exists a coanalytic zero-dimensional space that is not $\sigma$-homogeneous. Along the way, we will discuss a notion of hereditary rigidity. It is an open problem whether every analytic zero-dimensional space is $\sigma$-homogeneous. This is joint work with Zoltán Vidnyánszky. |