On Universal Mappings and Free Topological Groups

It has been observed that constructions so apparently different as Kronecker products, extension of the ring of operators of a module, field of quotients of an integral domain, free groups, free topological groups, completion of a uniform space, Cech compactification enter in the same frame. We intend in this paper to explain a rather general process of construction which may be applied to most of the examples quoted above. This paper will proceed axiomatically. In fact the problem under question (problem of a "universal mapping") can be only stated after a certain number of axioms. When the method of construction has been explained we shall illustrate it by the classical example of the completion of a uniform space. For more examples the reader is referred to a forthcoming book of N. Bourbaki. The same method gives also necessary and sufficient conditions for many imbedding problems. Both topological and algebraic examples will be given. In the last part of the paper our method of construction will be applied to MarkofFs theory of free topological groups.