Specker’s theorem for Nöbeling’s group

Abstract:

Specker proved that the group $\mathbb {Z}^{\aleph _0}$ of integer-valued sequences is far from free; all its homomorphisms to $\mathbb {Z}$ factor through finite subproducts. Nöbeling proved that the subgroup $\mathcal {B}$ consisting of the bounded sequences is free and therefore has many homomorphisms to $\mathbb {Z}$. We prove that all “reasonable” homomorphisms $\mathcal {B}\to \mathbb {Z}$ factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on $\mathcal {B}$. In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker’s theorem applies to $\mathcal {B}$ as well as to $\mathbb {Z}^{\aleph _0}$.