Page:Topological Groups-Deane Montgomery.djvu/8

 1. INTRODUCTION

Topological groups were first studied by Lie. The groups studied by him, called Lie groups, are the most important and most thoroughly studied type. Some of the other important contributors to the theory of Lie groups and topological groups are Hilbert, Brouwer, E. Cartan, H. Weyl, von Neumann, and Pontrjagin.

A topological group is a collection of elements which is a group and also a "space", of such a nature that the group operations are continuous (in a sense that is a direct generalization of continuity of functions of one or two real variables). The most familiar example is the set of real numbers, with addition as the group operation and the "space" as the real numbers with distance defined by d(x, y) = |x-y|. Other examples will be given later. In order to define topological groups more precisely, it is necessary to begin with a definition of "space". A rather general definition could be given. However, for these lectures it will be sufficient to use metric spaces. Hereafter, "space" will be understood to mean "metric space".

Definition. A set R of elements is called a metric space if for every two elements x and y in R there is associated a non-negative real number d(x, y), called the Rh