GENERALIZED TOPOLOGICAL OPERATOR (g-Tg-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES (Tg-SPACES): PART III. GENERALIZED DERIVED (g-Tg-DERIVED) AND GENERALIZED CODERIVED (g-Tg-CODERIVED) OPERATORS

Öz

In a T_g-space T_g = (Ω, T_g), the g-topology T_g : P (Ω) → P (Ω) can be characterized in the generalized sense by the novel g-T_g-derived, g-T_g-coderived operators g-Der_g, g-Cod_g : P (Ω) → P (Ω), respectively, giving rise to novel generalized g-topologies on Ω. In this paper, which forms the third part on the theory of g-T_g-operators in T_g-spaces, we study the essential properties of g-Der_g, g-Cod_g : P (Ω) → P (Ω) in T_g-spaces. We show that (g-Der_g, g-Cod_g) : P (Ω) × P (Ω) → P (Ω) × P (Ω) is a pair of both dual and monotone g-T_g-operators that is (∅, Ω), (∪, ∩)-preserving, and (⊆, ⊇)-preserving relative to g-T_g-(open, closed) sets. We also show that (g-Der_g, g-Cod_g) : P (Ω) × P (Ω) → P (Ω) × P (Ω) is a pair of weaker and stronger g-T_g-operators. Finally, we diagram various relationships amongst der_g, g-Der_g, cod_g, g-Cod_g : P (Ω) → P (Ω) and present a nice application to support the overall study.

Anahtar Kelimeler

• Generalized topological space (Tg-space)
• generalized sets (g-Tgsets)
• generalized derived operator (g-Tg-derived operator)
• generalized coderived operator (g-Tgcoderived operator)

Kaynakça

1. G. Cantor, Uber die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen, Math. Ann., Vol.5, pp.123-132 (1872).
2. G. Cantor, Uber Unendliche, Lineaere Punktmannigfaltigkeiten, Ibid., Vol.20, No.III, pp.113-121 (1882).
3. S. Ahmad, Absolutely Independent Axioms for the Derived Set Operator, The American Mathematical Monthly, Vol.73, No.4, pp.390-392 (1966).
4. A. Baltag and N. Bezhanishvili and A. Ozgun and S. Smets, A Topological Apprach to Full Belief, Journal of Philosophical Logic, Vol.48, No.2, pp.205-244 (2019).
5. M. Caldas and S. Jafari and M. M. Kov_ar, Some Properties of _-Open Sets, Divulgaciones Matem_aticas, Vol.12, No.2, pp.161-169 (2004).
6. D. Cenzer and D. Mauldin, On the Borel Class of the Derived Set Operator, Bull. Soc. Math. France, Vol.110, pp.357{380 (1982).
7. C. Devamanoharan and S. Missier and S. Jafari, _-Closed Sets in Topological Spaces, Eur. J. Pure Appl. Math., Vol.5, No.4, pp.554-566 (2012).
8. M. M. Fr_echet, Sur Quelques Points du Calcul Functionnel, Rend. Circ. Matem. Palermo, Vol.22, pp.1-72 (1906).

9. Y. Gnanambal, On Generalized Preregular Closed Sets in Topological Spaces, Indian J. Pure Appl. Math., Vol.28, pp.351-360 (1997).
10. F. Harary, A Very Independent Axiom System, Monthly, Vol.68, pp.159-162 (1961).
11. F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly, Vol.70, No.10, pp. 1085-1086 (1963).
12. E. R. Hedrick, On Properties of a Domain for Which Any Derived Set is Closed, Transactions of the American Mathematical Society, Vol.12, No. 3, pp.285-294 (1911)
13. D. Higgs, Iterating the Derived Set Function, The American Mathematical Monthly, Vol.90, No.10, pp.693-697 (1983).
14. H. J. Kowalsky, Topologische Raume, Birkhauser, Verlag, Basel und Stuttgart, pp.53 (1961).
15. R. M. Latif, Characterizations and Applications of -Open Sets, Soochow Journal of Mathematics, Vol.32, No.3, pp.1-10 (2006).
16. Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science, Vol.345, pp.63-76 (2019).
17. S. P. Missier and V. H. Raj, g__-Closed Sets in Topological Spaces, Int. J. Contemp. Math. Sciences, Vol.7, No.20, pp.963-974 (2012).
18. S. Modak, Some Points on Generalized Open Sets, Caspian Journal of Mathematical Sciences (CJMS), Vol.6, No. 2, pp.99-106 (2017).
19. G. H. Moore, The Emergence of Open Sets, Closed Sets, and Limit Points in Analysis and Topology, Historia Mathematica, Vol.35, No. 3, pp.220-241 (2008).
20. A. Al-Omari and M. S. M. Noorani, On b-Closed Sets, Bull. Malays. Sci. Soc., Vol.32, No.1, pp.19-30 (2009).
21. R. Rajendiran and M. Thamilselvan, Properties of g_s_-Closure, g_s_-Interior and g_s_Derived Sets in Topological Spaces, Applied Mathematical Sciences, Vol. 8, No. 140, pp.6969-6978 (2014).
22. J. A. R. Rodgio and J. Theodore and H. Jansi, Notions via __-Open Sets in Topological Spaces, IOSR Journal of Mathematics (IOSR-JM), Vol.6, No.3, pp. 25-29 (2013).
23. N. E. Rutt, On Derived Sets, National Mathematics Magazine, Vol. 18, No. 2, pp.53-63 (1943).
24. K. Sano and M. Ma, Alternative Semantics for Vissier's Propositional Logics, 10th International Tbilisi Symposium on Logic, Language, and Computation, TbiLLC 2013, LNCS 8984, Springer, pp.257-275 (2015).
25. C. Sekar and J. Rajakumari, A New Notion of Generalized Closed Sets in Topological Spaces, International Journal of Mathematics Trends and Technology (IJMTT), Vol. 36, No. 2, pp.124-129 (2016).
26. R. Spira, Derived-Set Axioms for Topological Spaces, Portugaliar Mathematica, Vol.26, pp.165-167 (1967).
27. C. Steinsvold, Topological Models of Belief Logics, Ph.D. Thesis, City University of New York (2007).
28. J. Tucker, Concerning Consecutive Derived Sets, The American Mathematical Monthly, Vol.74, No.5, pp.555-556 (1967).
29. J. Dixmier, General Topology, Springer Verlag New York Inc., Vol.18, pp.31-35 (1997).
30. S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, Vol.18, pp.31-35 (1970).
31. M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in General-ized Topological Spaces: Part II. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, (2023).
32. M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in General-ized Topological Spaces: Part I. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, (2023).
33. M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Connectedness (g-Tg-Connectedness) in Generalized Topological Spaces (Tg-Spaces), Journal of Universal Mathematics, Vol.6, No.1, pp.1-38 (2023).
34. M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties, Fundamentals of Contemporary Mathematical Sciences, Vol.3, No.2, pp.98-118 (2022).
35. M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties, Fundamentals of Contemporary Mathematical Sciences, Vol.3, No.1, pp.26-45 (2022).
36. M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics, Vol.5, No.1, pp.1-23 (2022).
37. M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory, Vol.36, pp.18-38 (2021).
38. M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R_eduit, Mauritius (2020).