A Precise Treatment of Soft Quotient Topology and Soft Covering Maps

Abstract

We develop a foundational theory of soft quotient topology, providing a systematic approach to quotient constructions in soft topological spaces. We establish the universal property of soft quotient topology and investigate the relationship between global soft topology and its parametric slices, noting that slice-wise quotient behaviour is not sufficient to characterise soft quotients. The usefulness of the framework is illustrated through the construction of soft quotient spaces, together with a study of soft group actions and their orbit spaces. We propose a precise definition of soft covering maps that resolves inconsistencies found in the existing literature. Finally, to illustrate the applicability of our framework, we discuss a potential application in multi-agent motion planning, showing how it can significantly reduce combinatorial complexity under parametric uncertainty.