Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Actual metric representing a fuzzy metric

  • 2 Accesses


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.

This is a preview of subscription content, log in to check access.


  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. (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.

Download references

Author information

Correspondence to Rajakumar Roopkumar.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Roopkumar, R., Vembu, R. Actual metric representing a fuzzy metric. J Anal (2020).

Download citation


  • Fuzzy metric
  • Metric
  • Topology

Mathematics Subject Classification

  • 54E35
  • 54A40
  • 26E50