Strictly Wider Class of Soft Sets via Supra Soft δ-Closure Operator

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DOI: https://doi.org/10.28924/2291-8639-22-2024-47
Published: Mar 11, 2024
Cite as:
Alaa M. Abd El-latif, Mesfer H. Alqahtani, F. A. Gharib, Strictly Wider Class of Soft Sets via Supra Soft δ-Closure Operator, Int. J. Anal. Appl., 22 (2024), 47.

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Alaa M. Abd El-latif, Mesfer H. Alqahtani, F. A. Gharib

Abstract

In this work, we use the supra soft δ-closure operator to present a new notion of generalized closed sets in supra soft topological spaces (or SSTSs), named supra soft δ-generalized closed sets. We show that, this notion is more general than many of previous notions, which presented before in famous papers. We illustrate many of its essential properties in detail. Specifically, we illustrate that the new collection neither forms soft topology nor supra soft topology. Moreover, we study the behavior of the soft image and soft pre-images of supra soft δ-generalized closed sets under new types of soft mappings, named supra soft irresolute and supra soft δ-irresolute closed. In addition, we define the concept of supra soft δ-generalized open sets, as a complement of supra soft δ-generalized closed sets. Finally, the relationships with other forms of generalized open sets in SSTSs are explored, supported by concrete examples and counterexamples. Therefore, I think the development of the notions presented in this paper are sufficiently general relevance to allow for future extensions.

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References

  1. A.S. Mashhour, A.A. Allam, F.S. Mahmoud, F.H. Khedr, On Supra Topological Spaces, Indian J. Pure Appl. Math. 4 (1983), 502–510.
  2. A. Alpers, Digital Topology: Regular Sets and Root Images of the Cross-Median Filter, J. Math. Imaging. Vis. 17 (2002), 7–14.
  3. A.M. Kozae, M. Shokry, M. Zidan, Supra Topologies for Digital Plane, AASCIT Commun. 3 (2016), 1–10.
  4. T.M. Al-shami, I. Alshammari, Rough Sets Models Inspired by Supra-Topology Structures, Artif. Intell. Rev. 56 (2022), 6855–6883. https://doi.org/10.1007/s10462-022-10346-7.
  5. A. Al-Omari, M.H. Alqahtani, Primal Structure with Closure Operators and Their Applications, Mathematics 11 (2023), 4946. https://doi.org/10.3390/math11244946.
  6. M.H. Alqahtani, New Topologies on a Graph, Adv. Appl. Discr. Math. 37 (2023), 77–93. https://doi.org/10.17654/0974165823013.
  7. M.H. Alqahtani, H.Y. Saleh, A Novel Class of Separation Axioms, Compactness, and Continuity via C-Open Sets, Mathematics. 11 (2023), 4729. https://doi.org/10.3390/math11234729.
  8. M.H. Alqahtani, F-open and F-closed sets in Topological Spaces, Eur. J. Pure Appl. Math. 16 (2023), 819–832. https://doi.org/10.29020/nybg.ejpam.v16i2.4583.
  9. A. Kharal, B. Ahmad, Mappings on Soft Classes, New Math. Nat. Comput. 07 (2011), 471–481. https://doi.org/10.1142/s1793005711002025.
  10. D. Molodtsov, Soft Set Theory-First Results, Comput. Math. Appl. 37 (1999), 19–31. https://doi.org/10.1016/s0898-1221(99)00056-5.
  11. P.K. Maji, R. Biswas, A.R. Roy, Soft Set Theory, Comput. Math. Appl. 45 (2003), 555–562. https://doi.org/10.1016/s0898-1221(03)00016-6.
  12. Z.A. Ameen, M.H. Alqahtani, Some Classes of Soft Functions Defined by Soft Open Sets Modulo Soft Sets of the First Category, Mathematics. 11 (2023), 4368. https://doi.org/10.3390/math11204368.
  13. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, γ-Operation and Decompositions of Some Forms of Soft Continuity in Soft Topological Spaces, Ann. Fuzzy Math. Inform. 7 (2014), 181–196.
  14. Z.A. Ameen, A Non-Continuous Soft Mapping That Preserves Some Structural Soft Sets, J. Intell. Fuzzy Syst. 42 (2022), 5839–5845. https://doi.org/10.3233/jifs-212410.
  15. I. Zorlutuna, H. akir, On Continuity of Soft Mappings, Appl. Math. Inf. Sci. 9 (2015), 403–409. https://doi.org/10.12785/amis/090147.
  16. D. Wardowski, On a Soft Mapping and Its Fixed Points, Fixed Point Theory Appl. 2013 (2013), 182. https://doi.org/10.1186/1687-1812-2013-182.
  17. Z.A. Ameen, M.H. Alqahtani, Baire Category Soft Sets and Their Symmetric Local Properties, Symmetry. 15 (2023), 1810. https://doi.org/10.3390/sym15101810.
  18. M.H. Alqahtani, Z.A. Ameen, Soft Nodec Spaces, AIMS Math. 9 (2024), 3289–3302. https://doi.org/10.3934/math.2024160.
  19. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, Soft Ideal Theory Soft Local Function and Generated Soft Topological Spaces, Appl. Math. Inf. Sci. 8 (2014), 1595–1603. https://doi.org/10.12785/amis/080413.
  20. A.H. Hussain, S.A. Abbas, A.M. Salman, N.A. Hussein, Semi Soft Local Function Which Generated a New Topology in Soft Ideal Spaces, J. Interdiscip. Math. 22 (2019), 1509–1517. https://doi.org/10.1080/09720502.2019.1706848.
  21. F. Gharib, A.M. Abd El-latif, Soft Semi Local Functions in Soft Ideal Topological Spaces, Eur. J. Pure Appl. Math. 12 (2019), 857–869.
  22. A.M. Abd El-Latif, Generalized Soft Rough Sets and Generated Soft Ideal Rough Topological Spaces, J. Intell. Fuzzy Syst. 34 (2018), 517–524. https://doi.org/10.3233/jifs-17610.
  23. M. Akdag, F. Erol, Soft I-Sets and Soft I-Continuity of Functions, Gazi Univ. J. Sci. 27 (2014), 923–932.
  24. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-Latif, γ-Operation and Decompositions of Some Forms of Soft Continuity of Soft Topological Spaces via Soft Ideal, Ann. Fuzzy Math. Inform. 9 (2015), 385–402.
  25. A. Nasef, M. Parimala, R. Jeevitha, M. ElSayed, Soft Ideal Theory and Applications, Int. J. Nonlinear Anal. Appl. 13 (2022), 1335–1342. https://doi.org/10.22075/ijnaa.2022.6266.
  26. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-Latif, Soft Regularity and Normality Based on Semi Open Soft Sets and Soft Ideals, Appl. Math. Inf. Sci. Lett. 3 (2015), 47–55.
  27. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, Soft Semi (Quasi) Hausdorff Spaces via Soft Ideals, South Asian J. Math. 4 (2014, 265–284.
  28. S.A. El-Sheikh, A.M. Abd El-latif, Decompositions of Some Types of Supra Soft Sets and Soft Continuity, Int. J. Math. Trends Technol. 9 (2014), 37–56. https://doi.org/10.14445/22315373/ijmtt-v9p504.
  29. M. Shabir, M. Naz, On Soft Topological Spaces, Comput. Math. Appl. 61 (2011), 1786–1799. https://doi.org/10.1016/j.camwa.2011.02.006.
  30. T.M. Al-shami, M.E. El-Shafei, Two Types of Separation Axioms on Supra Soft Topological Spaces, Demonstr. Math. 52 (2019), 147–165. https://doi.org/10.1515/dema-2019-0016.
  31. C.G. Aras, S. Bayramov, Results of Some Separation Axioms in Supra Soft Topological Spaces, TWMS J. Appl. Eng. Math. 9 (2019), 58–63.
  32. A.M. Abd El-Latif, R.A. Hosny, Supra Soft Separation Axioms and Supra Irresoluteness Based on Supra b-Open Soft Sets, Gazi Univ. J. Sci. 29 (2016), 845–854.
  33. A.M. Abd El-latif, S. Karata¸s, Supra b-Open Soft Sets and Supra b-Soft Continuity on Soft Topological Spaces, J. Math. Comput. Appl. Res. 5 (2015), 1–18.
  34. A.M. Abd El-latif, Soft Supra Strongly Generalized Closed Sets, J. Intell. Fuzzy Syst. 31 (2016), 1311–1317. https://doi.org/10.3233/ifs-162197.
  35. Z.G. Ergul, S. Yuksel, Supra Regular Generalized Closed Sets in Supra Soft Topological Spaces, Ann. Fuzzy Math. Inform. 11 (2016), 349–360.
  36. W. Rong, F. Lin, Soft Connected Spaces and Soft Paracompact Spaces, Int. J. Appl. Math. Stat. 51 (2013), 667–681.
  37. A.M. Abd El-latif, Supra Soft b-Connectedness I: Supra Soft b-Irresoluteness and Separateness, Creat. Math. Inform. 25 (2016), 127–134.
  38. A.M. Abd El-latif, Supra Soft b-Connectedness II: Some Types of Supra Soft b-Connectedness, Creat. Math. Inform. 26 (2017), 1–8.
  39. T.M. Al-shami, A. Mhemdi, R. Abu-Gdairi, M.E. El-Shafei, Compactness and Connectedness via the Class of Soft Somewhat Open Sets, AIMS Math. 8 (2023), 815–840. https://doi.org/10.3934/math.2023040.
  40. A.M. Abd El-latif, On Soft Supra Compactness in Supra Soft Topological Spaces, Tbilisi Math. J. 11 (2018), 169–178. https://doi.org/10.32513/tbilisi/1524276038.
  41. T. Al-shami, J. Alcantud, A. Azzam, Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms, Mathematics. 10 (2022), 4488. https://doi.org/10.3390/math10234488.
  42. C.G. Aras, S. Bayramov, Separation Axioms in Supra Soft Bitopological Spaces, Filomat. 32 (2018), 3479–3486. https://doi.org/10.2298/fil1810479g.
  43. A.M. Abd El-latif, R.A. Hosny, Supra Semi Open Soft Sets and Associated Soft Separation Axioms, Appl. Math. Inf. Sci. 10 (2016), 2207–2215. https://doi.org/10.18576/amis/100623.
  44. A.M. Abd El-latif, Supra Soft Separation Axioms Based on Supra β-Open Soft Sets, Math. Sci. Lett. 5 (2016), 121–129. https://doi.org/10.18576/msl/050202.
  45. A.M. Abd El-latif, Novel Types of Supra Soft Operators via Supra Soft sd-Sets and Applications, AIMS Math. 9 (2024), 6586–6602. https://doi.org/10.3934/math.2024321.
  46. A.M. Abd El-latif, M.H. Alqahtani, Novel Categories of Supra Soft Continuous Maps via New Soft Operators, AIMS Math. 9 (2024), 7449–7470. https://doi.org/10.3934/math.2024361.
  47. A.M. Abd El-latif, M.H. Alqahtani, New Soft Operators Related to Supra Soft δi -Open Sets and Applications, AIMS Math. 9 (2024), 3076–3096. https://doi.org/10.3934/math.2024150.
  48. K. Kannan, Soft Generalized Closed Sets in Soft Topological Spaces, J. Theor. Appl. Inf. Technol. 37 (2012), 17–21.
  49. K. Kannan, D. Rajalakshmi, R. Srikanth, Soft Strongly g-Closed Sets, Indian J. Sci. Technol. 8 (2015), 1–6. https://doi.org/10.17485/ijst/2015/v8i18/65394.
  50. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, Supra Generalized Closed Soft Sets with Respect to an Soft Ideal in Supra Soft Topological Spaces, Appl. Math. Inf. Sci. 8 (2014), 1731–1740. https://doi.org/10.12785/amis/080430.
  51. Z.A. Ameen, M.H. Alqahtani, Congruence Representations via Soft Ideals in Soft Topological Spaces, Axioms 12 (2023), 1015. https://doi.org/10.3390/axioms12111015.
  52. T.M. Al-shami, Improvement of the Approximations and Accuracy Measure of a Rough Set Using Somewhere Dense Sets, Soft Comput. 25 (2021), 14449–14460. https://doi.org/10.1007/s00500-021-06358-0.
  53. T.M. Al-shami, A. Mhemdi, Approximation Operators and Accuracy Measures of Rough Sets From an InfraTopology View, Soft Comput. 27 (2022), 1317–1330. https://doi.org/10.1007/s00500-022-07627-2.
  54. I. Zorlutuna, M. Akdag, W.K. Min, S. Atmaca, Remarks on Soft Topological Spaces, Ann. Fuzzy Math. Inform. 3 (2012), 171–185.
  55. L. Lincy, A. Kalaichelvi, Supra Soft Regular Open Sets, Supra Soft Regular Closed Sets and Supra Soft Regular Continuity, Int. J. Pure Appl. Math. 119 (2018), 1075–1079.
  56. A.M. Abd El-latif, R.A. Hosny, Supra Open Soft Sets and Associated Soft Separation Axioms, Int. J. Adv. Math. 6 (2017), 68–81.
  57. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, Soft Semi Compactness via Soft Ideals, Appl. Math. Inf. Sci. 8 (2014), 2297–2306.
  58. A. Kandil, O.A.E. Tantawy, S.A. El-Sheikh, A.M. Abd El-latif, Soft Semi Separation Axioms and Irresolute Soft Functions, Ann. Fuzzy Math. Inform. 8 (2014), 305–318.