Soft Uniform Spaces and Soft Uniform Continuity: Induced Topologies, Separation, and Compactness-Type Results

TL;DR

This paper develops the theory of soft uniform spaces, establishing their topological properties, continuity concepts, and compactness results, extending classical uniform space theory into a parameterized soft setting.

Contribution

It introduces the concept of soft uniformity, explores its induced topology, and proves a soft Heine--Cantor theorem, connecting soft compactness with soft completeness and total boundedness.

Findings

Soft uniformities induce soft T1 and regular topologies.

On soft compact domains, soft continuity implies soft uniform continuity.

Soft compactness ensures soft completeness and total boundedness.

Abstract

Soft uniform structures provide a way to speak about uniform closeness in a parameterized setting. Working over a fixed parameter set, we treat entourages as soft relations and introduce a notion of \emph{soft uniformity} whose axioms parallel the classical entourage approach. Every soft uniformity induces a canonical soft topology; moreover, the uniformity is separated exactly when the induced topology is soft $T_1$, and the induced topology is soft regular. We then study soft uniformly continuous mappings and prove a soft Heine--Cantor type theorem: on a soft compact domain, soft continuity already forces soft uniform continuity. Finally, soft total boundedness and soft completeness are formulated via soft Cauchy filters, and we show that soft compactness implies both properties. Examples are included to relate the theory to uniformities generated from classical structures and to highlight the role played by parameters.