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

Fu-gui Shi

doi:10.15672/hujms.1209995

Research Article

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

Year 2024, Volume: 53 Issue: 2, 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.

Year 2024, Volume: 53 Issue: 2, 457 - 470, 23.04.2024

Fu-gui Shi

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

Abstract

Keywords

The 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.

There are 21 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Fu-gui Shi 0000-0001-8090-3872
Project Number	This project was supported by the National Natural Science Foundation of China (11871097).
Early Pub Date	August 15, 2023
Publication Date	April 23, 2024
Published in Issue	Year 2024 Volume: 53 Issue: 2

Cite

APA	Shi, F.-g. (2024). Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line. Hacettepe Journal of Mathematics and Statistics, 53(2), 457-470. https://doi.org/10.15672/hujms.1209995
AMA	Shi Fg. Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):457-470. doi:10.15672/hujms.1209995
Chicago	Shi, Fu-gui. “Fuzzifying Pseudo-Quasi-Metric Topologies on the Fuzzy Real Line”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 457-70. https://doi.org/10.15672/hujms.1209995.
EndNote	Shi F-g (April 1, 2024) Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line. Hacettepe Journal of Mathematics and Statistics 53 2 457–470.
IEEE	F.-g. Shi, “Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 457–470, 2024, doi: 10.15672/hujms.1209995.
ISNAD	Shi, Fu-gui. “Fuzzifying Pseudo-Quasi-Metric Topologies on the Fuzzy Real Line”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 457-470. https://doi.org/10.15672/hujms.1209995.
JAMA	Shi F-g. Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line. Hacettepe Journal of Mathematics and Statistics. 2024;53:457–470.
MLA	Shi, Fu-gui. “Fuzzifying Pseudo-Quasi-Metric Topologies on the Fuzzy Real Line”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 457-70, doi:10.15672/hujms.1209995.
Vancouver	Shi F-g. Fuzzifying pseudo-quasi-metric topologies on the fuzzy real line. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):457-70.