Soft Metric Spaces via Soft Elements: Topology Across Parameter-Cardinalities and Fixed-Point Theory

Abstract

A soft set $(F,E)$ determines the selection space $\SE(F)=\prod_{e\in E}F(e)$. This paper studies two natural structures on that space. For at most countable $E$, the series metric $\dPi$ induces the product topology. For arbitrary $E$, the sup metric $\dsup$ induces uniform convergence. We prove that these metric spaces are complete exactly when all fibres are complete. We then compare global contractions with coordinatewise contractions and give counterexamples when coordinate separability or a uniform contractive bound is absent. Standard fixed-point theorems for complete metric spaces are recorded as direct consequences, without repeating their classical proofs. For uncountable $E$, the product topology may fail to be metrizable, so we work with its product uniformity. A parameterwise contraction theorem gives a unique fixed point and convergence in the product topology; a common bound below one gives convergence in $\dsup$.

Substantially revised after peer review. Elementary proofs of classical fixed-point theorems have been removed, the exposition has been shortened, assumptions and references have been corrected, and the treatment of uncountable parameter sets has been strengthened. 11 pages, 1 table