Aspects of automatic continuity

TL;DR

This paper explores automatic continuity in Polish groups, examining variants of the closed graph theorem, and establishing continuity results for homomorphisms in various mathematical contexts.

Contribution

It provides new proofs and generalizations of automatic continuity results for homomorphisms between Polish groups, including homeomorphism and isometry groups.

Findings

Automatic continuity holds for homeomorphism groups of compact manifolds.

Variants of the closed graph theorem are applicable to Polish groups.

The amount of choice needed for discontinuous homomorphisms is characterized.

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.