VII - Topological Groups. Applications

Abstract

This chapter presents the elementary, fundamental facts of the theory of topological groups and allied structures. Many of the spaces that appear in mathematics and its applications posess, besides their topological structure, an equally important algebraic structure. For instance, the real line R is also a field, while Rn is a vector space over the field of reals. When a set carries both a topological and an algebraic structure, one can expect nontrivial connections between these structures only if they are compatible with each other. The compatibility condition demands that all the algebraic operations be continuous. If the algebraic structure is that of a group, this procedure leads to the notion of topological group.