Coproducts of abelian topological groups

This paper is a study of certain topological group topologies on the weak or restricted direct sum of a family of abelian topological groups. Amongst the topologies studied are the topology of the coproduct in the category of abelian topological groups and Kaplanʹs asterisk topology. A simple, explicit description of the coproduct topology is given, and a detailed investigation is made of the relations between the coproduct topology, the asterisk topology and several other topologies of interest. The reflexivity properties of the topological coproduct are also analysed. In particular, it is shown that the coproduct of a family of locally compact Hausdorff abelian topological groups is reflexive if and only if all but countably many of the groups are discrete.