Every Polish group has a non-trivial topological group automorphism

Abstract

We prove that every Polish group with more than two elements admits a non-trivial topological group automorphism. As a consequence, a hypothetical uniquely homogeneous Polish space with more than two points cannot be a semitopological group.

Proposition 2.5 cannot be true in general since it implies disconnectedness, and there are connected Boolean Polish groups. The error comes from Axiom (O4) since it does not necessarily implies transitivy of the defined "order"