Specker's theorem for Nöbeling's group (Q2781322)

Let \(P=\mathbb{Z}^{\aleph_0}\) be the product of countably many copies of the integers and let \(B\) be the Abelian subgroup consisting of all bounded sequences. Specker proved in 1950 that the group \(P\) is far from being free by showing that \(P\) has only countably many homomorphisms into the ring of integers. In fact, Specker proved that every homomorphism \(h\colon P\to\mathbb{Z}\) factors through a truncation map (a map which truncates an infinite sequence after the first \(k\) terms for a fixed natural number \(k\)). In contrast to Specker's result Nöbeling proved in 1968 that \(B\) is a free group assuming the axiom of choice. In this paper the author proves that the axiom of choice was unavoidable. He shows that any homomorphism \(h\colon P\to\mathbb{Z}\) which has the Baire property restricted to \(\{0,1\}^{\aleph_0}\) factors through a truncation map. This implies in particular that it is consistent with the absense of the axiom of choice that \(B\) is as far from being free as \(P\). Further corollaries are obtained using axioms of set theory.