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 A N ( X ) (i.e., the free abelian group in the class of all nuclear groups) exists. We give two different descriptions of the topology on A N ( X ) . ards we study the question under which conditions the free abelian topological group A ( X ) is topologically isomorphic to the free nuclear group A N ( X ) and show that if X is a product of metrizable spaces then this is true if and only if X is discrete.