Actual metric representing a fuzzy metric

Abstract

For a given fuzzy metric M, we introduce two different nets $(\varDelta _{M,\lambda })$ and $(\delta _{M,\lambda })$ of metrics constructed from the fuzzy metric M, and prove that both nets converge to the same limit, under a necessary and sufficient condition. The common limit is called the actual metric representing the fuzzy metric M. We also derive some of the properties of these approximate metrics $\varDelta _{M,\lambda }$ and $\delta _{M,\lambda }$. On the other hand, for a given metric d, we 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 metric d induces a fuzzy metric $M_d$, then all the approximate metrics $\varDelta _{M,\lambda }$ and $\delta _{M,\lambda }$ constructed from this fuzzy metric are equal to the original metric d.