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

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### 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.

### Most cited references4

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### Author and article information

###### Journal
10 July 2019
###### Article
1907.05241