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