SavedAdd your journal

Journals / Journal of Universal Mathematics / 2026 / Cilt: 5 Sayı: 1

THEORY OF GENERALIZED SEPARATION AXIOMS IN GENERALIZED TOPOLOGICAL SPACES

Mohammad Irshad Khodabocus(University of Mauritius)0000-0003-2252-4342
Noor-ul-hacq Sookıa(University of Mauritius)0000-0002-3155-0473
Pages
1–23
Publication Date
2022-03-01
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms is defined in terms of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) and the properties and characterizations of a Tg-space endowed with each such g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms are discussed.
📊
Key Findings & Data:The study shows that g-Tg,F-axiom implies g-Tg,K-axiom, g-Tg,H-axiom implies g-Tg,F-axiom, g-Tg,R-axiom implies g-Tg,H-axiom, and g-Tg,N-axiom implies g-Tg,R-axiom.
💡
Conclusion & Impact:Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Generalized Topology · Generalized Topological Space · Generalized Separation Axioms

Abstract

In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms is defined in terms of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) and the properties and characterizations of a Tg-space endowed with each such g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms are discussed. The study shows that g-Tg,F-axiom implies g-Tg,K-axiom, g-Tg,H-axiom implies g-Tg,F-axiom, g-Tg,R-axiom implies g-Tg,H-axiom, and g-Tg,N-axiom implies g-Tg,R-axiom. Considering the Tg,K, Tg,F, Tg,H, Tg,R, Tg,N-axioms as their analogues but defined in terms of corresponding elements belonging to the class of open, closed, semi-open, semi-closed, preopen, preclosed, semi-preopen, and semi-preclosed sets, the study also shows that the statement Tg,α-axiom implies g-Tg,α-axiom holds for each α ∈ {K, F, H, R, N}. Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.

🔖

Cite This Article

Khodabocus, M. I., Sookıa, N. (2022). THEORY OF GENERALIZED SEPARATION AXIOMS IN GENERALIZED TOPOLOGICAL SPACES. Journal of Universal Mathematics, Cilt: 5 Sayı: 1, 1–23. https://doi.org/10.33773/jum.974278

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (27)

  1. W. K. Min, Remarks on Separation Axioms on Generalized Topological Spaces, Journal of the Chungcheong Mathematical Society, vol. 23, N. 2, pp. 293-298 (2010).
  2. D. Narasimhan, An Overview of Separation Axioms in Recent Research, International Journal of Pure and Applied Mathematics, vol. 76, N. 4, pp. 529-548 (2012). ISSN: 1311-8080 [3] B. Roy and R. Sen and T. Noiri, Separation Axioms on Topological Spaces: A Unified Version, European Journal of Pure and Applied Mathematics, vol. 6, N. 1, pp. 44-52 (2013). ISSN 1307-5543
  3. C. Carlos and R. Namegalesh and R. Ennis, Separation Axiom on Enlargements of Generalized Topologies, Revista Integración, vol. 32, N. 1, pp. 19-26 (2014).
  4. A. Danabalan and C. Santhi, A Class of Separation Axioms in Generalized Topology, Mathematical Journal of Interdisciplinary Sciences, vol. 4, N. 2, pp. 151-159 (2016).
  5. J. Dontchev, On Some Separation Axioms Associated with the α-Topology, Mem. Fac. Sci. Kochi Univ. Ser. A, Math., vol. 18, pp. 31-35 (1997).
  6. S. P. Missier and A. Robert, Higher Separation Axioms via Semi∗-Open Sets, Int. Journal of Engineering and Science, vol. 4, N. 6, pp. 37-45 (2014).
  7. T. C. K. Raman and V. Kumari and M. K. Sharma, α-Generalized and α∗-Separation Axioms for Topological Spaces, IOSR Journal of Mathematics, vol. 10, N. 3, pp. 32-36 (2014).
  8. F. G. Arenas and J. Dontchev and M. L. Puertas, Unification Approach to the Separation Axioms Between T0 and Completely Hausdorff, Acta Math. Hungar., vol. 86, N. 1-2, pp. 75-82 (2000).
  9. M. M. Deza and E. Deza, Encyclopedia of Distances, Discrete Mathematics, Springer-Verlag, Berlin, (2009). [11] K. El-Saady and F. Al-Nabbat, Separation Axioms in Generalized Base Spaces, British Journal of Mathematics and Computer Science, vol. 17, N. 2, pp. 1-8 (2016).
  10. R. Ennis and C. Carlos and S. José, γ-(α, β)-Semi Sets and New Generalized Separation Axioms, Bull. Malays. Sci. Soc. (2), vol. 30, N. 1, pp. 13-21 (2007).
  11. A. Keskin and T. Noiri, Higher Separation Axioms via Semi∗-Open Sets, Bulletin of the Iranian Mathematical Society, vol. 35, N. 1, pp. 179-198 (2009).
  12. V. Pankajam and D. Sivaraj, Some Separation Axioms in Generalized Topological Spaces, Bol. Soc. Paran. Mat., vol. 31, N. 1, pp. 29-42 (2013).
  13. D. Pratulananda and A. R. Mamun, g∗-Closed Sets and a New Separation Axiom in Alexandroff Spaces, Archivum Mathematicum (BRNO), vol. 39, pp. 299-307 (2003).
  14. L. A. Steen and J. A. Jr. Seebach, Counterexamples in Topology, Springer, New York, (1978).
  15. Á. Császár, Separations Axioms for Generalized Topologies, Acta Math. Hungar., vol. 104, N. 1-2, pp. 63-69 (1998).
  16. M. S. Sarsak, New Separation Axioms in Generalized Topological Spaces, Acta Math. Hungar., vol. 132, N. 3, pp. 244-252 (2011).
  17. L. L. L. Lusanta and H. M. Rara, Generalized Star α − b-Separation Axioms in Bigeneralized Topological Spaces, Applied Mathematical Sciences, vol. 9, N. 75, pp. 3725-3737 (2015).
  18. M. V. Mielke, Separation Axioms and Geometric Realizations, Indian J. Pure Appl. Math., vol. 25, N. 7, pp. 711-722 (1994).
  19. A. D. Ray and R. Bhowmick, Separation Axioms on Bi-Generalized Topological Spaces, Journal of the Chungcheong Mathematical Society, vol. 27, N. 3, pp. 363-379 (year).
  20. J. Thomas and S. J. John, Soft Generalized Separation Axioms in Soft Generalized Topological Spaces, International Journal of Scientific and Engineering Research, vol. 6, N. 3, pp. 969-974 (year).
  21. Á. Császár, Remarks on Quasi-Topologies, Acta Math. Hungar., vol. 119, N. 1-2, pp. 197-200 (2008).
  22. Á. Császár, Generalized Open Sets in Generalized Topologies, Acta Math. Hungar., vol. 106, N. 1-2, pp. 53-66 (2005).
  23. V. Pavlović and A. S. Cvetković, On Generalized Topologies arising from Mappings, Vesnik, vol. 38, N. 3, pp. 553-565 (year).
  24. S. Bayhan and A. Kanibir and I. L. Reilly, On Functions between Generalized Topological Spaces, Appl. Gen. Topol., vol. 14, N. 2, pp. 195-203 (2013).
  25. C. Boonpok, On Generalized Continuous Maps in Čech Closure Spaces, General Mathematics, vol. 19, N. 3, pp. 3-10 (2011).
  26. A. S. Mashhour and A. A. Allam and F. S. Mahmoud and F. H. Khedr, On Supratopological Spaces, Indian J. Pure. Appl. Math., vol. 14, N. 4, pp. 502-510 (1983).
  27. M. I. Khodabocus, A Generalized Topological Spaces endowed with Generalized Topologies, PhD Thesis, University of Mauritius, (2020).