I've been told that every topological abelian group is a product of Eilenberg-Mac Lane spaces, but I don't have a reference for this fact. This confuses me because via the Dold-Kan correspondence, this should mean that every (nonnegatively-graded) chain complex of abelian groups is quasi isomorphic to a complex with zero differentials, ie that every complex is formal. But this is not the case.
So what am I missing? Maybe the weak equivalence between a topological abelian group and a product of Eilenberg-Mac Lane spaces need not commute with the abelian group structure? But then where did this map come from?