Automatic continuity of abstract homomorphisms between locally compact and Polish groups

TL;DR

This paper investigates conditions under which abstract group homomorphisms between certain classes of topological groups, such as locally compact and Polish groups, are automatically continuous and open, extending classical results to broader contexts.

Contribution

The authors develop an axiomatic framework that generalizes and unifies automatic continuity results across various classes of topological groups.

Findings

Established criteria for automatic continuity in broad classes of groups

Unified approach applicable to locally compact and Polish groups

Extended classical automatic continuity results to new settings

Abstract

We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $\cal K$ of spaces, and suppose that $\phi:K \to G$ is an abstract (i.e. not necessarily continuous) surjective group homomorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $\phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting.