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

TL;DR

This paper introduces and explores the concepts of soft connectedness and soft path connectedness within soft topological groups, establishing their properties, relationships, and categorical structure, including the formation of a symmetric monoidal category.

Contribution

It defines soft topologies compatible with usual topologies, introduces soft paths, and constructs the category of soft topological groups as a symmetric monoidal category.

Findings

Soft connectedness and soft path connectedness are preserved under soft continuous maps.

The category of soft topological groups is symmetric monoidal.

Soft topologies on $ ext{R}$ and $[0,1]$ are developed.

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.