Generalized Kuratowski Closure Operators in the Bipolar Metric Setting

Abstract

We initiate the investigation of the topological aspects of bipolar metric spaces. In this context, some concepts that generalize open and closed balls and sets, accumulation points, closure and interior operators for bipolar metric spaces, of which little is known about their topological behaviors, are discussed. In addition, some essential properties regarding these notions were obtained, and counterexamples were provided for some expected but not satisfied properties.

Keywords

• closure operator
• bipolar metric
• open set

References

1. H. Ahmad, M. Younis, A. A. N. Abdou, Bipolar b−metric spaces in graph setting and related fixed points, Symmetry, 15(6) (2023), 1227–1242.
2. B. Alamri, Fixed point results in F−bipolar metric spaces with applications, Mathematics, 11(10) (2023), 2399–2412.
3. D. Bajovi´c, Z. D. Mitrovi´c, M. Saha, Remark on contraction principle in conetvs b−metric spaces, The Journal of Analysis, 29(1) (2021), 273–280.
4. M. Baran, Closure operators in convergence spaces, Acta Mathematica Hungarica, 87(1-2) (2000), 33–45.
5. A. Bartwal, R. C. Dimri, G. Prasad, Some fixed point theorems in fuzzy bipolar metric spaces, Journal of Nonlinear Sciences and Applications 13 (2020), 196– 204.
6. S. Bayramov, Ç. Aras Gündüz, H. Poşul, A study on bipolar soft metric spaces, Filomat, 37(10) (2023) 3217–3224.
7. I. Beg, A. J. Gnanaprakasam, Common coupled fixed point theorem on fuzzy bipolar metric spaces, International Journal of Nonlinear Analysis and Applications, 14(4) (2023), 77–85.
8. M. Caldas, S. Jafari, R. M. Latif, A. A. Nasef, On a closure space via semi-closure operator, Turkish Journal of Computer and Mathematics Education, 12(14) (2021), 5447–5445.

9. G. D. Çaylı, Generating nullnorms on some special classes of bounded lattices via closure and interior operators, Information Sciences, 552 (2021), 118–141.
10. S. Çetin, U. Gürdal, Characterization of bipolar ultrametric spaces and some fixed point theorems, Hacettepe Journal of Mathematics and Statistics, 52(1) (2023), 185-196.
11. A. Erciyes, T. M. Baran, M. Qasim, Closure operators in constant filter convergence spaces, Konuralp Journal of Mathematics, 8(1) (2020), 185–191.
12. M. M. Fr´echet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, 22(1) (1906), 1–72.
13. Y. U. Gaba, M. Aphane, V. Sihag, On two Banach-type fixed points in bipolar metric spaces, Abstract and Applied Analysis, 2021 (1), 4846877.
14. U. Gürdal, C¸ ift kutuplu metrik uzaylar ve sabit nokta teoremleri, Doctoral dissertation, PhD Thesis. Manisa Celˆal Bayar University, Graduate School of Natural and Applied Sciences, (2018).
15. U. Gürdal A. Mutlu, K. Özkan, Fixed point results for αψ−contractive mappings in bipolar metric spaces, Journal of Inequalities & Special Functions, 11(1) (2020), 64–75.
16. S. Gurusamy, Bicomplex valued bipolar metric spaces and fixed point theorems, Mathematical Analysis and its Contemporary Applications, 4(1) (2022), 29–43.
17. P. Hitzler, A. K. Seda, Dislocated topologies, Journal of Electrical Engineering 51(12) (2000), 3–7.
18. D. Holgate, J. Slapal, Closure, interior and neighbourhood in a category, Hacettepe Journal of Mathematics and Statistics, 47(6) (2018), 1512–1520.
19. J. K. Kim, M. Kumar, ω−interpolative contractions in bipolar metric space, Nonlinear Functional Analysis and Applications, 28(2) (2023), 383–394.
20. J. K. Kim, M. Kumar, ψ−type contraction and Jaggi type hybrid contraction in bipolar metric spaces, Nonlinear Functional Analysis and Applications, 28(3) (2023), 703–717.
21. G. N. V. Kishore, R. P. Agarwal, B. Srinuvasa Rao, R. V. N. Srinivasa Rao, Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications, Fixed point theory and Applications, 2018(1) (2018), 1–13.
22. M. Kumar, P. Kumar, A. Mutlu, R. Ramaswamy, Q. A. A. Abdelnaby, S. Radenovi´c, Ulam–Hyers stability and well-posedness of fixed point problems in C∗−algebra valued bipolar b−metric spaces, Mathematics, 11(10) (2023), 2323–2336.
23. M. Kumar, P. Kumar, R. Ramaswamy, Q. A. A. Abdelnaby, A. Elsonbaty, S. Radenovi´c, (α−ψ) Meir–Keeler contractions in bipolar metric spaces, Mathematics, 11(6) (2023), 1310–1323.
24. S. Kumar, M. Sharma, A fixed point result in bipolar metric spaces, Neuroquantology, 20(17) (2022), 312– 315.
25. A. Malhotra, D. Kumar, Bipolar R−metric space and fixed point result, International Journal of Nonlinear Analysis and Applications, 13(2) (2022), 709–712.
26. G. Mani, A. J. Gnanaprakasam, Z. D. Mitrovi´c, M. E. Bota, Solving an integral equation via fuzzy triple controlled bipolar metric spaces, Mathematics, 9(24) (2021), 3181–3194.
27. G. Mani, A. J. Gnanaprakasam, H. Is¸ık, F. Jarad, Fixed point results in C⋆-algebra-valued bipolar metric spaces with an application, AIMS Mathematics, 8(4) (2023), 7695–7713.
28. G. Mani, A. J. Gnanaprakasam, P. Subbarayan, S. Chinnachamy, R. George, Z. D. Mitrovi´c, Applications to nonlinear fractional differential equations via common fixed point on C⋆−algebra-valued bipolar metric spaces, Fractal and Fractional, 7(7) (2023), 534–555.
29. G. Mani, R. Ramaswamy, A. J. Gnanaprakasam, A. Elsonbaty, Q. A. A. Abdelnaby, S. Radenovi´c, Application of fixed points in bipolar controlled metric space to solve fractional differential equation, Fractal and Fractional 7 (3) (2023), 242–261.
30. G. Mani, A. J. Gnanaprakasam, S. Kumar, Ö. Ege, M. De la Sen, Fixed-point theorems for nonlinear contraction in fuzzy-controlled bipolar metric spaces, Axioms, 12(4) (2023), 396–407.
31. P. P. Murthy, C. P. Dhuri, S. Kumar, R. Ramaswamy, M. A. S. Alaskar, S. Radenovi´c, Common fixed point for Meir–Keeler type contraction in bipolar metric space, Fractal and Fractional, 6(11) (2022), 649–660.
32. A. Mutlu, U. Gürdal, Bipolar metric spaces and some fixed point theorems, Journal of Nonlinear Sciences and Applications, 9(9) (2016), 5362–5373.
33. A. Mutlu, K. Özkan, U. Gürdal Coupled fixed point theorems on bipolar metric spaces, European journal of Pure and Applied Mathematics, 10(4) (2017), 655–667.
34. A. Mutlu, K. Özkan, U. Gürdal, A new fixed point theorem in modular metric spaces, International Journal of Analysis and Applications, 16(4) (2018), 472–483.
35. A. Mutlu, K. Özkan, U. Gürdal, Fixed point theorems for multivalued mappings on bipolar metric spaces, Fixed Point Theory, 21(1) (2020), 271–280.
36. A. Mutlu, K. Özkan, U. Gürdal, Locally and weakly contractive principle in bipolar metric spaces, TWMS Journal of Applied and Engineering Mathematics, 10(2) (2020), 379–388.
37. K. Özkan, U. Gürdall, The fixed point theoremand characterization of bipolarmetric completeness, Konuralp Journal of Mathematics, 8(1) (2020), 137–143.
38. S. Özkan, M. Kula, S. Kula, T. M. Baran, Closure operators, irreducibility, Urysohn’s lemma, and Tietze extension theorem for proximity spaces, Turkish Journal of Mathematics, 47(2) (2023), 870–882.
39. N. Parkala, U. R. Gujjula, S. R. Bagathi, Some quadruple fixed points of integral type contraction mappings in bipolar metric spaces with application, Indian Journal of Science and Technology, 16(25) (2023), 1843– 1856.
40. J. L. Pfaltz, Closure operators associated with networks, Hacettepe Journal of Mathematics and Statistics, 46(1) (2017), 91–101.
41. M. Qasim, M. Baran, H. Abughalwa, Closure operators in convergence approach spaces, Turkish Journal of Mathematics, 45(1) (2021), 139–152.
42. B. Ramalingam, Ö . Ege, A. Aloqaily, N. Mlaiki, Fixed-point theorems on fuzzy bipolar b−metric spaces, Symmetry, 15(1831) (2023), 1–17.
43. B. S. Rao, E. Gouthami, C. Maheswari, M. S. Shaik, Coupled coincidence points for hybrid pair of mappings via mixed monotone property in bipolar metric spaces, Advances in Fixed Point Theory, 12(3) (2022), 1–17.
44. G. U. Reddy, C. Ushabhavani, B. S. Rao, Existence fixed point solutions for C−class functions in bipolar metric spaces with applications, International Journal of Analysis and Applications, 21(7) (2023), 1–17.
45. M. Şahin, A. Kargın, M. S. Uz, A. Kılıc¸, Neutrosophic triplet bipolar metric spaces, Neutrosophic Sets and Systems, 33 (2020), 297–313.
46. M. Sangurlu Sezen, Some special functions in orthogonal fuzzy bipolar metric spaces and their fixed point applications, Numerical Methods for Partial Differential Equations, 38(4) (2022), 794–802.
47. W. A. Wilson, On quasi-metric spaces, American Journal of Mathematics, 53(3) (1931), 675–684.