Zero-dimensional $\sigma$-homogeneous spaces

TL;DR

This paper investigates the conditions under which zero-dimensional separable metrizable spaces are $\sigma$-homogeneous, revealing that set-theoretic assumptions significantly influence this property and introducing new concepts of hereditary rigidity.

Contribution

It establishes the impact of different set-theoretic axioms on $\sigma$-homogeneity of zero-dimensional spaces and introduces hereditary rigidity notions.

Findings

Under $ extsf{AD}$, all zero-dimensional spaces are $\sigma$-homogeneous.

Under $ extsf{AC}$, some zero-dimensional spaces are not $\sigma$-homogeneous.

Under $ extsf{V=L}$, there exist non-$\sigma$-homogeneous coanalytic zero-dimensional spaces.

Abstract

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 obtain the following results: assuming $\mathsf{AD}$, every zero-dimensional space is $\sigma$-homogeneous; assuming $\mathsf{AC}$, there exists a zero-dimensional space that is not $\sigma$-homogeneous; assuming $\mathsf{V=L}$, there exists a coanalytic zero-dimensional space that is not $\sigma$-homogeneous. Along the way, we introduce two notions of hereditary rigidity, and give alternative proofs of results of van Engelen, Miller and Steel. It is an open problem whether every analytic zero-dimensional space is $\sigma$-homogeneous.