2026/06/14 by Subhasis Ray · 1 voice
Mathematics · #math.GN
A soft set (F,E) determines the selection space \SE(F)=∏e∈ EF(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.