ABSTRACT
The study of the symmetries of geometric objects has interested generations of mathematicians for hundreds of years. A symmetry of a geometric object is merely a self-equivalence of the object, and the groups (in the algebraic sense) are intended to analyse symmetries of such objects. In most instances, groups arise as a family of continuous transformations acting on a space. These groups of continuous transformations receive topological structures, in a natural way. The mathematical structures, in which the notions of group and topology are blended, are called \Topological Groups." Their origin can be traced in Klein's (1872) programme to study geometries through transformation groups associated with them, and in the work of Lie (1873) on \continuous groups." However, the abstract notion of a topological group was introduced by O. Schreier (1925) and F. Leja (1927). These partly geometric objects form a rich territory of interesting examples in topology and geometry, due to the presence of the two basic interrelated mathematical structures in one and the same set. In this section, we see some important examples and the basic properties of topological groups.
