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.
"Automatic continuity for homomorphisms into free products." J. Symbolic Logic 78 (4) 1288 - 1306, December 2013. https://doi.org/10.2178/jsl.7804160