Some remarks on metrics induced by a fuzzy metric

We introduce a crisp metric $d_M$ as the common limit of two different nets $({\mathrm{\Delta }}_{M,\lambda })$ 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 ${\mathrm{\Delta }}_{\lambda }$ 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 ${\mathrm{\Delta }}_{M,\lambda }$ and ${\delta }_{M,\lambda }$ induced by this fuzzy metric are equal to the original metric $d$.