Free products of topological groups

Abstract

In this note the notion of a free topological product G α of a set { G α } of topological groups is introduced. It is shown that it always exists, is unique and is algebraically isomorphic to the usual free product of the underlying groups. Further if each G α is Hausdorff, then G α is Hausdorff and each G α is a closed subgroup. Also G α is a free topological group (respectively, maximally almost periodic) if each G α is. This notion is then combined with the theory of varieties of topological groups developed by the author. For a variety of topological groups, the -product of groups in is defined. It is shown that the -product, G α of any set { G α } of groups in exists, is unique and is algebraically isomorphic to the usual varietal product. It is noted that the -product of Hausdorff groups is not necessarily Hausdorff, but is if is abelian. Each G α is a quotient group of G α . It is proved that the -product of free topological groups of and projective topological groups of are of the same type. Finally it is shown that G α is connected if and only if each G α is connected.