Automatic continuity for homomorphisms into free products

TL;DR

This paper proves that any homomorphism from a completely metrizable topological group into a free product of groups is continuous unless its image is contained in a factor, implying all such topologies are discrete.

Contribution

It establishes a general automatic continuity result for homomorphisms into free products, extending understanding of topological group structures.

Findings

Homomorphisms into free products are continuous unless their image lies in a factor.

Any completely metrizable topology on a free product must be discrete.

The result applies broadly to topological groups and free product structures.

Abstract

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.