TOPOLOGIES ON THE FUNCTION SPACE \(Y^X\) WITH VALUES IN A TOPOLOGICAL GROUP

Abstract

Let \(Y^X\) denote the set of all functions from \(X\) to \(Y\). When \(Y\) is a topological space, various topologies can be defined on \(Y^X\). In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ generalized open sets in the topological space \(Y\). This approach also applies to the set of all continuous functions from \(X\) to \(Y\), denoted by \(C(X,Y)\), particularly when \(Y\) is a topological group. In investigating various topologies on both \(Y^X\) and \(C(X,Y)\), the concept of limit points plays a crucial role. The notion of a topological ideal provides a useful tool for defining limit points in such spaces. Thus, we utilize topological ideals to study the properties and consequences for function spaces and topological groups.

Keywords

• topological group, topological ideal, function space \(y^x\)