An Open Mapping Theorem

TL;DR

The paper proves that any surjective group homomorphism from an arbitrary power of integers to a locally compact group is open, resolving a previously posed mathematical question.

Contribution

It establishes a general open mapping theorem for surjective morphisms from infinite product groups to locally compact groups, extending classical results.

Findings

Surjective morphisms from $\mathbb{Z}^\kappa$ to locally compact groups are open.

Answers a longstanding open problem in topological group theory.

Generalizes the open mapping theorem to infinite product groups.

Abstract

It is proved that any surjective morphism $f: \mathbb{Z}^\kappa \to K$ onto a locally compact group $K$ is open for every cardinal $\kappa$. This answers a question posed by Karl Heinrich Hofmann and the second author.