TITLE: Every free abelian group is slender, why?
QUESTION [7 upvotes]: Wikipedia states that every free abelian group is slender. Where can I find a proof?
If this is not trivial, then I will also need a reference to use in my paper.

REPLY [8 votes]: This follows fairly straightforwardly from the fact that every map from $A=\mathbb{Z}^\mathbb{N}$ to $\mathbb{Z}$ factors through a finite subproduct (let me call this "Specker's theorem").  Suppose $F$ is a free abelian group and $f:A\to F$ is a homomorphism.  Since a subgroup of a free abelian group is free, we may assume $f$ is surjective.  If $F$ is finitely generated, the conclusion follows from Specker's theorem, so we may assume $F$ is infinitely generated.  Since $F$ is free, the map $f$ splits and so $F$ is a direct summand of $A$.  But there are uncountably many homomorphisms $F\to\mathbb{Z}$, and hence uncountably many homomorphisms $A\to \mathbb{Z}$.  This contradicts Specker's theorem.