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

TL;DR

This work extends classical topology to the soft setting by defining the soft usual topology on $\mathbb{R}$ and introducing soft path concepts on the subspace $I=[0,1]$. It establishes fundamental results for soft connectedness and soft path connectedness, including their preservation under soft continuous mappings and their behavior in soft topological groups. The paper then constructs the category of soft topological groups, proves its categorical structure, and shows it forms a symmetric monoidal category with a soft product, terminal object, and projections. Together, these contributions provide a parameterized framework for studying topological and algebraic properties under uncertainty and lay groundwork for a soft analogue of fundamental group theory.

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.

Theorems & Definitions (91)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Definition 2.6
• Example 2.7
• Proposition 2.8
• Definition 2.9
• Definition 2.10