The relationship between pointwise $k$-pseudo metrics and pointwise $k$-closed ball systems

Abstract

In this paper, we introduce a definition of a pointwise $k$-pseudo metric which is a generalization of pointwise pseudo metric. Then we discuss the axiomatic characterizations of a pointwise $k$-pseudo metric by a pointwise $k$-closed ball system that can be regarded as the opposite of the corresponding pointwise $k$-open ball system. Moreover, it is proved that there is a one-to-one correspondence between pointwise $k$-pseudo metrics and symmetric pointwise $k$-closed ball systems, and the symmetry conditions are different and simpler than the previous symmetry conditions. Finally, some $L$-structures induced by a pointwise $k$-closed ball system are obtained, including an $L$-neighborhood system,  an $L$-interior operator and an $L$-topology ($L$ denotes a completely distributive De Morgan algebra).

Keywords

• pointwise k-pseudo metric
• pointwise k-closed ball system
• L-neighborhood system
• L-interior operator
• L-topology

References

1. [1] I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal. 30, 26–37, 1989.
2. [2] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24, 182–190, 1968.
3. [3] S. Czerwik, Contraction mapping in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1, 5–11, 1993.
4. [4] Z. K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86, 74–95, 1982.
5. [5] P. Dwinger, Characterizations of the complete homomorphic images of a completely distributive complete lattice I, Indag. Math. (Proc.) 85, 403–414, 1982.
6. [6] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69, 205–230, 1979.
7. [7] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst. 90, 365–368, 1997.
8. [8] G. Gierz, K. H. Hofmann and K. Keimel, A Compendium of Continuous Lattices, Springer, Berlin, 1980.

9. [9] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18, 145–174, 1967.
10. [10] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27, 385–389, 1988.
11. [11] J. Gutiérrez García and M. A. de Prada Vicente, Hutton $[0,1]$-quasi-uniformities induced by fuzzy (quasi-)metric spaces, Fuzzy Sets Syst. 157, 755–766, 2006.
12. [12] X. Han and B. Pang, Categorical and convergent properties of convex spaces, Filomat 39 (4), 1163–1179, 2025.
13. [13] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12, 215–229, 1984.
14. [14] I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11, 336–344, 1975.
15. [15] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, Adam Mickiewicz University, Poznan, Poland, 1985.
16. [16] T. Kubiak and A. P. Šostak, A fuzzification of the category of M-valued L-topological spaces, Appl. Gen. Topol. 5, 137–154, 2004.
17. [17] J. H. Liang, Some questions on fuzzy metrics, Ann. Math. 1, 59–67, 1984.
18. [18] Y. M. Liu and M. K. Luo, Fuzzy Topology, World Scientific, Singapore, 1998.
19. [19] N. N. Morsi, On fuzzy pseudo-normed vector spaces, Fuzzy Sets Syst. 27, 351–372, 1988.
20. [20] B. Pang, Convergence structures in M-fuzzifying convex spaces, Quaest. Math. 43 (11), 1541–1561, 2020.
21. [21] B. Pang, Bases and subbases in (L,M)-fuzzy convex spaces, Comput. Appl. Math. 39, 41, 2020.
22. [22] B. Pang and W. Yao, The Scott open V -set monad and its categories of algebras, Commun. Algebra 53 (11), 4887–4913, 2025.
23. [23] Y. W. Peng, Pointwise p.q. metrics and its induced the family of mappings on completely distributive lattices, Ann. Math. 3, 353–359, 1992.
24. [24] C. Shen, Y. Shi, F. G. Shi and H. Andradi, Characterizations of pointwise pseudometrics via pointwise closed-ball systems, IEEE Trans. Fuzzy Syst. 30 (5), 1212–1223, 2022.
25. [25] F. G. Shi, Pointwise pseudo-metric in L-fuzzy set theory, Fuzzy Sets Syst. 121, 209– 216, 2001.
26. [26] F. G. Shi, L-fuzzy interiors and L-fuzzy closures, Fuzzy Sets Syst. 160, 1218–1232, 2009.
27. [27] F. G. Shi, (L,M)-fuzzy metric spaces, Indian J. Math. 52, 231–250, 2010.
28. [28] F. G. Shi and Z. Y. Xiu, A new approach to the fuzzification of convex structures, J. Appl. Math. 2014, 1–12, 2014.
29. [29] F. G. Shi and Z. Y. Xiu, (L,M)-fuzzy convex structures, J. Nonlinear Sci. Appl. 10 (7), 3655–3669, 2017.
30. [30] A. P. Šostak, On a fuzzy topological structure, Suppl. Rend. Circ. Mat. Palermo Ser. II 11, 89–103, 1985.
31. [31] A. P. Šostak, Some remarks on fuzzy k-pseudometric spaces, Filomat 32, 3567–3580, 2018.
32. [32] L. Sun and B. Pang, Lattice representation of L-quasi-convex spaces, Hacet. J. Math. Stat., doi:10.15672/hujms.1741223, 2026.
33. [33] H. Yang and B. Pang, Fuzzy points based betweenness relations in L-convex spaces, Filomat 35 (10), 3521–3532, 2021.
34. [34] Y. Yue and F. G. Shi, On fuzzy pseudo-metric spaces, Fuzzy Sets Syst. 161, 1105– 1116, 2010.
35. [35] L. A. Zadeh, Fuzzy sets, Inf. Control 8, 338–353, 1965.
36. [36] Y. Zhong, Y. H. Guo and Z. H. Yang, Research on k-norms and fuzzy k-norms, Comput. Appl. Math. 44, 112, 2025.
37. [37] Y. Zhong and S. Lin, M-fuzzifying k-pseudo-metric space and its induced M-fuzzifying structures, Comput. Appl. Math. 41, 41–70, 2022.
38. [38] Y. Zhong, S. Lin and J. Chang, Characterizations of M-fuzzifying topological operators and its applications, Filomat 39 (3), 905–919, 2025.
39. [39] Y. Zhong and F. G. Shi, Characterizations of (L,M)-fuzzy topological degrees, Iran. J. Fuzzy Syst. 15 (4), 129–149, 2018.
40. [40] Y. Zhong and F. G. Shi, Stratified (L,M)-fuzzy derived spaces, Iran. J. Fuzzy Syst. 15 (4), 109–127, 2018.
41. [41] Y. Zhong and A. P. Šostak, A new definition of fuzzy k-pseudo-metric and its induced fuzzifying structures, Iran. J. Fuzzy Syst. 18, 55–66, 2021.
42. [42] Y. Zhong, A. P. Šostak and F. G. Shi, Pointwise k-pseudo-metric space, Mathematics 9, 2505, 2021.
43. [43] Y. Zhong, X. Wu, A. P. Šostak and F. G. Shi, (L,M)-fuzzy k-pseudo-metric space, Mathematics 10, 1151, 2022.