chapter  3
26 Pages

Topological Groups

WithL. S. Pontryagin

From a purely logical point of view a topological group is obtained by uniting the concepts of group and of topological space; it is simply assumed that in one and the same set G there is defined an operation of group multiplication turning G into a group, as well as an operation of topological closure turning G into a topological space. These operations, however, are not independent but are connected by a continuity condition: the group operations in G are required to be continuous in the topological space G. Such being the definition, it is not surprising that the first few steps in the development of the theory of topological groups disclose almost nothing that is specific to it. The basic facts and concepts pertinent to groups, and to topological spaces are simply translated more or less immediately into the context of topological groups. Thus we encounter subgroups, normal subgroups, factor groups, and so forth. To be sure, certain situations specifically pertaining to topological groups do turn up in the process but they are comparatively superficial. It is to the study of these quite general properties of topological groups that the present chapter is devoted. A deeper investigation of topological groups will be given in subsequent chapters.