Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

Abstract

Schweizer, Sklar and Thorp proved in 1960 that a Menger space \((G,D,T)\) under a continuous \(t\)-norm \(T\), induce a natural topology \(\tau\) wich is metrizable. We extend this result to any probabilistic metric space \((G,D,\star)\) provided that the triangle function \(\star\) is continuous. We prove in this case, that the topological space \((G,\tau)\) is uniformly homeomorphic to a (deterministic) metric space \((G,\sigma_D)\) for some canonical metric \(\sigma_D\) on \(G\). As applications, we extend the fixed point theorem of Hicks to probabilistic metric spaces which are not necessarily Menger spaces and we prove a probabilistic Arzela-Ascoli type theorem.