Aspects of automatic continuity
Abstract
A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and improvements of the closed graph theorem, applying these to the problem of continuity of universally measurable homomorphisms and also to gauge the amount of choice needed to construct discontinuous homomorphisms between Polish groups. Furthermore, we provide a simple proof of automatic continuity in the context of homeomorphism groups of compact manifolds and a complete reworking of automatic continuity theory in the context of isometry groups of highly homogeneous complete metric structures.
Cite
@article{arxiv.2406.12143,
title = {Aspects of automatic continuity},
author = {Christian Rosendal and Luis Carlos Suarez},
journal= {arXiv preprint arXiv:2406.12143},
year = {2025}
}
Comments
We introduce several new results regarding automatic continuity in models of AC+DC without Vitali sets, which involves novel applications of the characteristic subgroup associated with a homomorphism. A mistake in the proof of Lemma 45 (Lemma 57 in the new version) has been corrected