| Résume | We show that a universally measurable homomorphism between Polish
groups is automatically continuous. Using our general analysis of
continuity of group homomorphisms, this result is used to calibrate the
strength of the existence of a discontinuous homomorphism between Polish
groups. In particular, it is shown that, modulo ZF+DC, the existence of
a discontinuous homomorphism between Polish groups implies that the
Hamming graph on {0, 1}N has finite chromatic number. This solves a
classical problem originating in JPR Christensen's work on Haar null sets. |
