On the probabilistic metrizability of approach spaces

TL;DR

This paper characterizes when approach spaces derived from probabilistic metric spaces are metrizable under different continuous t-norms, linking probabilistic metrizability to specific t-norm properties.

Contribution

It establishes a criterion for probabilistic metrizability of approach spaces based on the supremum of idempotent elements of the t-norm.

Findings

Probabilistic metrization aligns with the minimum t-norm when $k^*=1$.

Probabilistic metrization aligns with the product t-norm when $k^*<1$.

Provides a clear condition linking t-norm properties to approach space metrizability.

Abstract

We investigate approach spaces generated by probabilistic metric spaces with respect to a continuous t-norm $*$ on the unit interval $[0,1]$. Let $k^*$ be the supremum of the idempotent elements of $*$ in $[0,1)$. It is shown that if $k^*=1$ (resp. $k^*<1$), then an approach space is probabilistic metrizable with respect to $*$ if and only if it is probabilistic metrizable with respect to the minimum (resp. product) t-norm.