On topological quotient hyperrings and α^*-relation

In this research, we first introduce the concept of a topological Krasner hyperring and then proceed to investigate its properties. By applying relative topology to subhyperrings, we analyze the properties associated with them. In other words, the aim is to utilize specific topologies to identify the diverse substructural characteristics of this type of hyperring. Additionally, we examine the quotient topology resulting from an interesting relation on the discussed spaces to understand how this relation influences the topological structure of the hyperring. Finally, we demonstrate that the topological Krasner hyperring induced by τ_α, which is the finest and strongest topology on ℋ, ultimately forms a ring. In summary, this research not only analyzes the structural properties of these hyperrings but also examines, from a topological perspective, how different relations impact this structure, proving that the resulting topology is strong enough to form a ring.