# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Zero-dimensional $\sigma$-homogeneous spaces"

#### Medini, Andrea

All 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.

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