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Dergiler / Proceedings of International Mathematical Sciences / 2026 / Volume: 5 Issue: 1

Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure

Mohammad Irshad Khodabocus(Université des Mascareignes)0000-0003-2252-4342
Noor-ul-hacq Sookıa(University of Mauritius)0000-0002-3155-0473
Sayfa
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Yayın Tarihi
2023-07-18
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The elements supporting these facts are reported therein as sources of inspiration for more generalized operations.

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Amaç & Hipotez:In a generalized topological space Tg = (Ω, Tg ) (Tg -space), various ordinary topological operators (Tg -operators), namely, int_g, cl_g, ext_g, fr_g, der_g, cod_g : P (Ω) −→ P (Ω) (T_g-interior, T_g-closure, T_g-exterior, T_g-frontier, T_g-derived, T_g-coderived operators), are defined in terms of ordinary sets (T_g-sets).
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Temel Bulgular & Veri:Accordingly, generalized T_g-operators (g-T_g-operators), namely, g-Int_g, g-Cl_g, g-Ext_g, g-Fr_g, g-Der_g, g-Cod_g : P (Ω) −→ P (Ω) (g-T_g-interior, g-T_g-closure, g-T_g-exterior, g-T_g-frontier, g-T_g-derived, g-T_g-coderived operators) may be defined in terms of generalized T_g-sets (g-T_g-sets), thereby making g-T_g-operators theory in T_g-spaces an interesting subject of inquiry.
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Sonuç & Katkı:The elements supporting these facts are reported therein as sources of inspiration for more generalized operations.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Generalized topological space · generalized sets · generalized interior operator

Abstract

In a generalized topological space Tg = (Ω, Tg ) (Tg -space), various ordinary topological operators (Tg -operators), namely, int_g, cl_g, ext_g, fr_g, der_g, cod_g : P (Ω) −→ P (Ω) (T_g-interior, T_g-closure, T_g-exterior, T_g-frontier, T_g-derived, T_g-coderived operators), are defined in terms of ordinary sets (T_g-sets). Accordingly, generalized T_g-operators (g-T_g-operators), namely, g-Int_g, g-Cl_g, g-Ext_g, g-Fr_g, g-Der_g, g-Cod_g : P (Ω) −→ P (Ω) (g-T_g-interior, g-T_g-closure, g-T_g-exterior, g-T_g-frontier, g-T_g-derived, g-T_g-coderived operators) may be defined in terms of generalized T_g-sets (g-T_g-sets), thereby making g-T_g-operators theory in T_g-spaces an interesting subject of inquiry. In this paper, we present the definitions and the essential properties of the g-T_g-interior and g-T_g-closure operators g-Int_g , g-Cl_g : P (Ω) −→ P (Ω), respectively, in terms of a new class of g-T_g-sets which we studied earlier. The outstanding results to which the study has led to are: Firstly, (g-Int_g, g-Cl_g) : P (Ω) × P (Ω) −→ P (Ω) × P (Ω) is (Ω, ∅)-grounded, (expansive, non-expansive), (idempotent, idempotent) and (∩, ∪)-additive. Secondly, g-Int_g : P (Ω) −→ P (Ω) is finer (or, larger, stronger than int_g : P (Ω) −→ P (Ω) and g-Cl_g : P (Ω) −→ P (Ω) is coarser (or, smaller, weaker) than cl_g : P (Ω) −→ P (Ω). The elements supporting these facts are reported therein as sources of inspiration for more generalized operations.

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Bu Makaleye Atıf Yap

Khodabocus, M. I., Sookıa, N. (2023). Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure. Proceedings of International Mathematical Sciences, Volume: 5 Issue: 1. https://doi.org/10.47086/pims.1214055

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Kaynakça (47)

  1. M. Caldas and S. Jafari and M. M. Kov´ar, Some Properties of θ-Open Sets, Divulgaciones Matem´aticas 12(2) (2004) 161-169.
  2. ´A. Cs´asz´ar, Further Remarks on the Formula for γ-Interior, Acta Math. Hungar., 113(4) (2006) 325-332.
  3. ´A. Cs´asz´ar, Generalized Open Sets in Generalized Topologies, Acta Math. Hungar. 106(1-2) (2005) 53-66.
  4. ´A. Cs´asz´ar, On the γ-Interior and γ-Closure of a Set, Acta Math. Hungar. 80 (1998) 89-93.
  5. ´A. Cs´asz´ar, Generalized Open Sets, Acta Math. Hungar. 75(1-2) (1997) 65-87.
  6. A. Gupta and R. D. Sarma, A Note on some Generalized Closure and Interior Operators in a Topological Space, Math. Appl. 6 (2017) 11-20.
  7. S. -M. Jung and D. Nam, Some Properties of Interior and Closure in General Topology, Mathematics (MDPI Journal) 7(624) (2019) 1-10.
  8. N. Kalaivani, Operation Approaches on α-β-Open Sets in Topological Spaces, Int. Journal of Math. Analysis 7(10) (2013) 491-498.
  9. N. Levine, Generalized Closed Set in Topological Spaces, Rend. Circ. Mat. Palermo 19 (1970) 89-96.
  10. N. Levine, Semi-Open Sets and Semi-Continuity in Topological Spaces, Amer. Math. Monthly 70 (1963) 19-41.
  11. N. Levine, On the Commutivity of the Closure and Interior Operators in Topological Spaces, Amer. Math. Monthly 68(5) (1961) 474-477.
  12. T. S. I. Mary and A. Gowri, The Role of q-Sets in Topology, International Journal of Mathematics Research 8(1) (2016) 1-10.
  13. W. Dungthaisong and C. Boonpok, Generalized Closed Sets in Bigeneralized Topological Spaces, Int. Journal of Math. Analysis 5(24) (2011) 1175-1184.
  14. A. Gupta and R. V. Sarma, PS-Regular Sets in Topology and Generalized Topology, Journal of Mathematics 2014(1-6) (2014) 1-6.
  15. M. I. Khodabocus and N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory 36 (2021) 18-38.
  16. M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R´eduit, Mauritius (2020) 1-311 (i.-xxxvi.).
  17. W. K. Min and Y. K. Kim, Quasi Generalized Open Sets and Quasi Generalized Continuity on Bigeneralized Topological Spaces, Honam Mathematical J. 32(4) (2010) 619-624.
  18. W. K. Min, Some Results on Generalized Topological Spaces and Generalized Systems, Acta. Math. Hungar. 108(1-2) (2005) 171-181.
  19. J. M. Mustafa, On Binary Generalized Topological Spaces, General Letters in Mathematics 2(3) (2017) 111-116.
  20. D. Andrijevi´c, On b-Open Sets, Mat. Vesnik 48 (1996) 59-64.
  21. J. Dixmier, General Topology, Springer Verlag New York Inc. 1 (1984) X-141.
  22. O. Nj˚astad, On Some Classes of Nearly Open Sets, Pacific J. of Math. 15(3) (1965) 961-970.
  23. S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts 18 (1970) 369.
  24. A. Al-Omari and M. S. M. Noorani, On b-Closed Sets, Bull. Malays. Sci. Soc. 32(1) (2009) 19-30.
  25. J. Dontchev and H. Maki, On θ-Generalized Closed Sets, Internat. J. Math. & Math. Sci. 22(2) (1999) 239-249.
  26. C. Kuratowski, Sur l’Op´eration A¯ de l’Analyse Situs, Fund. Math. 3 (1922) 182-199.
  27. J. F. Z. Camargo, Some Properties of Beta Hat Generalized Closed Set in Generalized Topological Spaces, International Journal for Research in Mathematics and Statistics 5(3) (2019) 1-8.
  28. W. K. Min, Mixed θ-Continuity on Generalized Topological Spaces, Mathematical and Computer Modelling 54(11-12) (2011) 2597-2601.
  29. V. Pankajam, On the Properties of δ-Interior and δ-Closure in Generalized Topological Spaces, International Journal for Research in Mathematical Archive 2(8) (2011) 1321-1332.
  30. D. Saravanakumar and N. Kalaivani and G. S. S. Krishnan, On ˜μ-Open Sets in Generalized Topological Spaces, Malaya J. Mat. 3(3) (2015) 268-276.
  31. B. K. Tyagi and R. Choudhary, On Generalized Closure Operators in Generalized Topological Spaces, International Journal of Computer Applications 82(15) (2013) 1-5.
  32. W. K. Min, A Note on θ (g, g′)-Continuity in Generalized Topological Spaces, Acta. Math. Hungar. 125(4) (2009) 387-393.
  33. W. K. Min, Mixed Weak Continuity on Generalized Topological Spaces, Acta. Math. Hungar. 132(4) (2011) 339-347.
  34. C. Cao and J. Yan and W. Wang and B. Wang, Some Generalized Continuities Functions on Generalized Topological Spaces, Hacettepe Journal of Mathematics and Statistics 42(2) (2013) 159-163.
  35. S. Srija and D. Jayanthi, gu-Semi Closed Sets in Generalized Topological Spaces, International Journal of Scientific Engineering and Applied Science (IJSEAS) 2(4) (2016) 292-294.
  36. C. Boonpok, (ζ, δ (μ))-Closed Sets in Strong Generalized Topological Spaces, Cogent Mathematics & Statistics 5(1517428) (2018) 1-45.
  37. K. Kannan and N. Nagaveni, On ˆβ-Generalized Closed Sets and Open Sets in Topological Spaces, Int. Journal of Math. Analysis 6(57) (2012) 2819-2828.
  38. ´A. Cs´asz´ar, Generalized Topology, Generalized Continuity, Acta Math. Hungar. 96(4) (2002) 351-357.
  39. V. Pavlovi´c and A. S. Cvetkovi´c, On Generalized Topologies arising from Mappings, Vesnik 38(3) (2012) 553-565.
  40. ´A. Cs´asz´ar, Remarks on Quasi-Topologies, Acta Math. Hungar. 119(1-2) (2008) 197-200.
  41. S. Bayhan and A. Kanibir and I. L. Reilly, On Functions between Generalized Topological Spaces, Appl. Gen. Topol. 14(2) (2013) 195-203.
  42. C. Boonpok, On Generalized Continuous Maps in ˇ Cech Closure Spaces, General Mathematics 19(3) (2011) 3-10.
  43. A. S. Mashhour and A. A. Allam and F. S. Mahmoud and F. H. Khedr, On Supratopological Spaces, Indian J. Pure. Appl. Math. 14(4) (1983) 502-510.
  44. M. Caldas and S. Jafari and R. K. Saraf, Semi-θ-Open Sets and New Classes of Maps, Bulletin of the Iranian Mathematical Society 31(2) (2005) 37-52.
  45. J. Dontchev, On Some Separation Axioms Associated with the α-Topology, 18 (1997) 31-35.
  46. Y. B. Jun and S. W. Jeong and H. J. Lee and J. W. Lee, Applications of Pre-Open Sets, Applied General Topology, Universidad Polit´ecnica de Valencia 9(2) (2008) 213-228.
  47. On Generalized Closed Sets in a Generalized Topological Spaces, CUBO A Mathematical Journal 18(1) (2016) 27-45.