Free nuclear groups

We define free groups in certain classes of abelian topological groups. Of special interest is the class of nuclear groups. For a completely regular Hausdorff space X we show that the free nuclear group AN(X) (i.e., the free abelian group in the class of all nuclear groups) exists. We give two different descriptions of the topology on AN(X).Afterwards we study the question under which conditions the free abelian topological group A(X) is topologically isomorphic to the free nuclear group AN(X) and show that if X is a product of metrizable spaces then this is true if and only if X is discrete.