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Actual metric representing a fuzzy metric

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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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References

  1. 1.

    Castro-Company, F., S. Romaguera, and P. Tirado. 2015. On the construction of metrics from fuzzy metrics and its application to the fixed point theory of multivalued mappings. Fixed Point Theory and Applications. https://doi.org/10.1186/s13663-015-0476-1. (Article ID. 226).

  2. 2.

    Chakrabarty, K., R. Biswas, and S. Nanda. 1998. On fuzzy metric spaces. Fuzzy Sets and Systems 99: 111–114.

  3. 3.

    Das, R.K. 2001. Fuzzy topology generated by fuzzy metrics. Indian Journal Of Pure and Applied Mathematics 32: 1899–1904.

  4. 4.

    Deng, Z. 1982. Fuzzy pseudo-metric spaces. Journal of Mathematical Analysis and Applications 86: 74–95.

  5. 5.

    Erceg, M.A. 1979. Metric spaces in fuzzy set theory. Journal of Mathematical Analysis and Applications 69: 205–230.

  6. 6.

    George, A., and P. Veeramani. 1994. On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64: 395–399.

  7. 7.

    Grabiec, M. 1989. Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 27: 385–389.

  8. 8.

    Gregori, V., and S. Romaguera. 2000. Some properties of fuzzy metric spaces. Fuzzy Sets and Systems 115: 485–489.

  9. 9.

    Kaleva, O., and S. Seikkala. 1984. On fuzzy metric spaces. Fuzzy Sets and Systems 12: 215–229.

  10. 10.

    Karamosil, I., and J. Michálek. 1975. Fuzzy metrics and statistical metric spaces. Kybernetika 11: 336–344.

  11. 11.

    Mahalanobis, P.C. 1936. On the generalized distance in statistics. Proceedings of the National Academy of Sciences, India, Section A 2: 49–55.

  12. 12.

    Menger, K. 1942. Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America 28: 535–537.

  13. 13.

    Menger, K., B. Schweizer, and A. Sklar. 1959. On probabilistic metrics and numerical metrics with probability 1. Czechoslovak Mathematical Journal 9: 459–466.

  14. 14.

    Miheţ, D. 2004. A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets and Systems 144: 431–439.

  15. 15.

    Nishiura, E. 1970. Constructive methods in probabilistic metric spaces. Fundamenta Mathematicae 67: 115–124.

  16. 16.

    Radu, V. 2002. Some remarks on the probabilistic contractions on fuzzy Menger spaces. Automation, Computers, Applied Mathematics 11: 125–131.

  17. 17.

    Radu, V. 2004. Some suitable metrics on fuzzy metric spaces. Fixed Point Theory 5: 323–347.

  18. 18.

    Roldán, A., J.M.- Moreno, and C. Roldán. 2013. On interrelationships between fuzzy metric structures. Iranian Journal of Fuzzy Systems 10: 133–150.

  19. 19.

    Savchenko, A., and M. Zarichnyi. 2009. Fuzzy ultrametrics on the set of probability measures. Topology 48: 130–136.

  20. 20.

    Schweizer, B., and A. Sklar. 1960. Statistical metric spaces. Pacific Journal of Mathematics 10: 313–334.

  21. 21.

    Wald, A. 1943. On a statistical generalization of metric spaces. Proceedings of the National Academy of Sciences of the United States of America 29: 196–197. USA.

  22. 22.

    Xia, Z.Q.-, and F.F.- Guo. 2004. Fuzzy metric spaces. Journal of Applied Mathematics and Computing 16: 371–381.

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Correspondence to Rajakumar Roopkumar.

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Roopkumar, R., Vembu, R. Actual metric representing a fuzzy metric. J Anal (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