2026/02/28 by S. Ray
Mathematics · #math.GN
Soft uniform structures provide a way to describe uniform closeness in a parameterized setting. Over a fixed parameter set, entourages are treated as soft relations, and a soft uniformity is defined by axioms parallel to the classical entourage axioms. Each soft uniformity induces two related topologies: a sectionwise soft topology \(τ\sU\) on \(F\), and an ordinary soft-element topology \(\TU\) on \(\SE(F)\). Both constructions are described. The space \((\SE(F),\TU)\) is Hausdorff, equivalently \(T1\), exactly when the soft uniformity is separated, and it is regular for every soft uniformity. Soft uniformly continuous mappings are then studied, and a Heine--Cantor type theorem is proved when the soft-element topology of the domain is compact. Finally, total boundedness and completeness are formulated for the canonical soft-element uniform space \((\SE(F),\sU\SE)\), and compactness of \((\SE(F),\TU)\) is shown to imply both properties. Examples relate the theory to uniformities generated by classical structures and illustrate the role of the parameter set.