Specker's theorem for Nöbeling's group

Specker proved that the group of integer-valued sequences is far from free; all its homomorphisms to factor through finite subproducts. Nöbeling proved that the subgroup consisting of the bounded sequences is free and therefore has many homomorphisms to . We prove that all ``reasonable' homomorphisms factor through finite subproducts. Among the reasonable homomorphisms are all those that are Borel with respect to a natural topology on . In the absence of the axiom of choice, it is consistent that all homomorphisms are reasonable and therefore that Specker's theorem applies to as well as to .