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.

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Correspondence to Rajakumar Roopkumar.

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Roopkumar, R., Vembu, R. Actual metric representing a fuzzy metric. J Anal 28, 973–985 (2020). https://doi.org/10.1007/s41478-020-00228-y

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Keywords

  • Fuzzy metric
  • Metric
  • Topology

Mathematics Subject Classification

  • 54E35
  • 54A40
  • 26E50