All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. In this talk, we will discuss the following results, which were inspired by Ostrovsky's theorem: (1) Assuming AD, every zero-dimensional space is σ-homogeneous, (2) Assuming AC, there exists a zero-dimensional space that is not σ-homogeneous, (3) Assuming V=L, there exists a coanalytic zero-dimensional space that is not σ-homogeneous. Along the way, we will 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 σ-homogeneous. This is joint work with Zoltán Vidnyánszky.