Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem - Archive ouverte HAL Access content directly
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## Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem

Mohammed Bachir
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Bruno Nazaret

#### 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.

#### Domains

Mathematics [math] Functional Analysis [math.FA]
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### Dates and versions

hal-02112322 , version 1 (26-04-2019)
hal-02112322 , version 2 (07-07-2019)

### Identifiers

• HAL Id : hal-02112322 , version 2
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### Cite

Mohammed Bachir, Bruno Nazaret. Metrization of probabilistic metric spaces. Applications to fixed point theory and Arzela-Ascoli type theorem. 2019. ⟨hal-02112322v2⟩

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