Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line

Year 2024, , 457 - 470, 23.04.2024

Fu-gui Shi

https://doi.org/10.15672/hujms.1209995

Abstract

In this paper, three natural fuzzifying topologies are presented on the fuzzy real line. Then the notion of fuzzifying pseudo-quasi-metrics is introduced. It is proved that the three fuzzifying topologies can be induced respectively by three fuzzifying pseudo-quasi-metrics. Our definition of fuzzifying pseudo-metric is slightly different from that of KM-fuzzy metric. A fuzzifying pseudo-metrics can be regarded as a weak form of a KM fuzzy metric.

Keywords

fuzzy real line, fuzzifying topology, fuzzifying pseudo-quasi-metric, fuzzy pseudo-metric

Project Number

This project was supported by the National Natural Science Foundation of China (11871097).

References

• [1] K. Bekar, Metric on $L$-fuzzy real line, International J. Math. Combin. 3, 48–60, 2022.
• [2] T.E. Gantner, R.C. Steinlage and R.H. Warren, Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62, 547–562, 1978.
• [3] R. Goetschel and W. Voxman, Topological properties of fuzzy numbers, Fuzzy Sets Syst. 10, 87–99, 1983.
• [4] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27, 385–389, 1988.
• [5] U. Höhle, Probabilistsche Metriken auf der Menge nicht negativen verteilungs funktionen, Aequationes Math. 18, 345–356, 1978.
• [6] H.-L. Huang and F.-G. Shi, $L$-fuzzy numbers and their properties, Inform. Sci. 178, 1141–1151, 2008.
• [7] B. Hutton, Normality in fuzzy topological spaces, J. Math. Anal. Appl. 50, 74–79, 1975.
• [8] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11, 326–334, 1975.
• [9] Y.-M. Liu and M.-K. Luo, Fuzzy topology, World Scientific, Singapore, 1997.
• [10] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10, 314–334, 1960.
• [11] F.-G. Shi, Pointwise uniformities in fuzzy set theory, Fuzzy Sets Syst. 98, 141–146, 1998.
• [12] F.-G. Shi, Pointwise metrics in fuzzy set theory, Fuzzy Sets Syst. 121, 209–216, 2001.
• [13] F.-G. Shi, Pointwise pseudo-metric on the $L$-real line, Iranian J. Fuzzy Syst. 2, 15–20, 2005.
• [14] F.-G. Shi, C.-Y. Zheng, Metrization theorems in $L$-topological spaces, Fuzzy Sets Syst. 149, 455–471, 2005.
• [15] F.-G. Shi, A new approach to $L$-$T_2$, $L$-Urysohn, and L-completely Hausdorff axioms, Fuzzy Sets Syst. 157, 794–803, 2006.
• [16] F.-G. Shi, $(L,M)$-fuzzy metric spaces, Indian J. Math. 52, 231–250, 2010.
• [17] F.-G. Shi and Z.-Y. Xiu, A new approach to the fuzzification of convex spaces, J. Appl. Math. 2014, 249183, 2014.
• [18] F.-G. Shi and Z.-Y. Xiu, $(L,M)$-fuzzy convex structures, J. Nonlinear Sci. Appl. 10, 3655–3669, 2017.
• [19] F.-G. Shi, L-metric on the space of $L$-fuzzy numbers, Fuzzy Sets Syst. 399, 95–109, 2020.
• [20] M. S. Ying, Fuzzifying topology based on complete residuated lattice-valued logic (I), Fuzzy Sets Syst. 56, 337–373, 1993.
• [21] D. Zhang, A natural topology for fuzzy numbers, J. Math. Anal. Appl. 264, 344–353, 2001.