Medini, A., & Vidnyánszky, Z. (2024). Zero-dimensional σ-homogeneous spaces. Annals of Pure and Applied Logic, 175(1), Article 103331. https://doi.org/10.1016/j.apal.2023.103331
All spaces are assumed to be separable and metrizable. Ostrovsky showed that every zero-dimensional Borel space is σ-homogeneous. Inspired by this theorem, we obtain the following results:
• Assuming 𝗔𝗗, every zero-dimensional space is σ-homogeneous,
• Assuming 𝗔𝗖, there exists a zero-dimensional space that is not σ-homogeneous,
• Assuming 𝗩 = 𝗟, there exists a coanalytic zero-dimensional space that is not σ-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 σ-homogeneous.
