THEORY OF GENERALIZED SEPARATION AXIOMS IN GENERALIZED TOPOLOGICAL SPACES

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.

Keywords

• Generalized Topology
• Generalized Topological Space
• Generalized Separation Axioms
• Generalized Sets

References

1. [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. [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. [4] 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. [5] 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. [6] 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. [7] 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. [8] 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. [9] 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. [10] 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. [12] 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. [13] 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. [14] 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. [15] 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. [16] L. A. Steen and J. A. Jr. Seebach, Counterexamples in Topology, Springer, New York, (1978).
15. [17] Á. Császár, Separations Axioms for Generalized Topologies, Acta Math. Hungar., vol. 104, N. 1-2, pp. 63-69 (1998).
16. [18] M. S. Sarsak, New Separation Axioms in Generalized Topological Spaces, Acta Math. Hungar., vol. 132, N. 3, pp. 244-252 (2011).
17. [19] 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. [20] M. V. Mielke, Separation Axioms and Geometric Realizations, Indian J. Pure Appl. Math., vol. 25, N. 7, pp. 711-722 (1994).
19. [21] 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. [22] 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. [23] Á. Császár, Remarks on Quasi-Topologies, Acta Math. Hungar., vol. 119, N. 1-2, pp. 197-200 (2008).
22. [24] Á. Császár, Generalized Open Sets in Generalized Topologies, Acta Math. Hungar., vol. 106, N. 1-2, pp. 53-66 (2005).
23. [25] V. Pavlović and A. S. Cvetković, On Generalized Topologies arising from Mappings, Vesnik, vol. 38, N. 3, pp. 553-565 (year).
24. [26] 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. [27] C. Boonpok, On Generalized Continuous Maps in Čech Closure Spaces, General Mathematics, vol. 19, N. 3, pp. 3-10 (2011).
26. [28] 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. [29] M. I. Khodabocus, A Generalized Topological Spaces endowed with Generalized Topologies, PhD Thesis, University of Mauritius, (2020).