Actual metric representing a fuzzy metric

For a given fuzzy metric M, we introduce two different nets (Delta 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 Delta M,lambda and delta M,lambda. On the other hand, for a given metric d, we establish that the fuzzy metric representing Md 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 Md, then all the approximate metrics Delta M,lambda and delta M,lambda constructed from this fuzzy metric are equal to the original metric d.