Some remarks on metrics induced by a fuzzy metric

TL;DR

The paper investigates how fuzzy metrics induce sequences of crisp metrics, establishing conditions for their convergence and compatibility with the original metric and topology.

Contribution

It introduces a new crisp metric as a limit of fuzzy metric-induced nets and characterizes its existence and properties, linking fuzzy and crisp metrics.

Findings

Existence of limits characterized by conditions on the fuzzy metric

Approximate metrics converge to the original metric under certain conditions

Fuzzy metrics with values in {0,1} are compatible with the same topology as the original metric

Abstract

We introduce a crisp metric $d_M$ as the common limit of two different nets $(\Delta_{M,\lambda})$ and $(\delta_{M,\lambda})$ of crisp metrics induced by a fuzzy metric $M$ and prove that the existence of each of these limits is equivalent to that of the other and it is characterized by another condition on the original fuzzy metric $M$. We also derive some of the properties of these approximate metrics $\Delta_\lambda$ and $\delta_\lambda$. On the other hand, for a given a crisp metric $d$, establish that the fuzzy metric representing $M_d$ with values in $\{0,1\}$ and $d$ are compatible with the same topology. Further, we prove that if a crisp metric $d$ induces a fuzzy metric $M_d$, then all the approximate crisp metrics $\Delta_{M,\lambda}$ and $\delta_{M,\lambda}$ induced by this fuzzy metric are equal to the original metric $d$.