Fundamental System and Boundary Structure of Topological Krasner Hypermodules

In this article, we first define hyperstructures known as Krasner hypermodules. Then, the concept of topological Krasner hypermodules is explored, examining their fundamental propertiesand the notion of continuous mappings that exist between such topological hyperstructures. Next, the concept of Hausdorff topology is introduced and its relation to Krasner hypermodules is examined. The relationship between locally compact Krasner hypermodules and the role of open neighborhoods in their topological structure is then analyzed. Several theorems are presented and proven to clarify these relationships. By applying relative topology to subhypermodules, their associated properties are analyzed. In other words, the aim is to use specific topologies to identify the various substructural features of this type of hypermodule. Additionally, the quotient topology induced by the θ^*-relation on the Krasner hypermodule is investigated to understand how this relation affects the topological structure of the hypermodule. Finally, it is shown that the topological Krasner hypermodule induced by τ_θ​, the finest and strongest topology on it, ultimately forms a module.