Soft Connectedness, Soft Path Connectedness and the Category of Soft Topological Groups

Abstract

In this study, the soft usual topology compatible with the usual topology of $\mathbb{R}$ is defined, and using its subspace topology on the interval $[0,1]$, the concept of a soft path is introduced. Within this context, the notions of soft connectedness and soft path connectedness are developed, their relationship is analyzed, and it is shown that these properties are preserved under soft continuous mappings. Moreover, the behavior of these concepts within soft topological groups is investigated in detail. Finally, the category of soft topological groups is constructed, its morphisms are identified, and it is shown that this category forms a symmetric monoidal category.