The Katowice problem and autohomeomorphisms of ω0⁎

We show that the existence of a homeomorphism between ω0⁎ and ω1⁎ entails the existence of a non-trivial autohomeomorphism of ω0⁎. This answers Problem 441 in [8].We also discuss the joint consistency of various consequences of ω0⁎ and ω1⁎ being homeomorphic.