On the Pettis-Johnstone theorem for localic groups

TL;DR

This paper interprets Johnstone's proof of the closed subgroup theorem for localic groups as a point-free analogue of Pettis's theorem, extending classical results to the localic setting and exploring their implications.

Contribution

It provides a novel point-free perspective on classical theorems for localic groups, including the open mapping theorem and automatic continuity, linking topology and locale theory.

Findings

Derived localic versions of the open mapping theorem

Established automatic continuity of Borel homomorphisms in the localic context

Proved the non-existence of binary coproducts of complete Boolean algebras

Abstract

We explain how Johnstone's 1989 proof of the closed subgroup theorem for localic groups can be viewed as a point-free version of Pettis's theorem for Baire topological groups. We then use it to derive localic versions of the open mapping theorem and automatic continuity of Borel homomorphisms, as well as the non-existence of binary coproducts of complete Boolean algebras.