Topologies on the Function Space \ (Yˣ\) With Values in a Topological Group

Let \ (YX\) denote the set of all functions from \ (X\) to \ (Y\). When \ (Y\) is a topological space, various topologies can be defined on \ (YX\). 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 \ (YX\) 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.