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

In this study, the soft usual topology compatible with the usual topology of 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.