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

Year 2023, , 183 - 220, 31.07.2023

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa , Radhakhrishna Dinesh Somanah

https://doi.org/10.33773/jum.1295736

Cited By: 1

Abstract

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.

Keywords

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

References

• G. Cantor, Uber die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen, Math. Ann., Vol.5, pp.123-132 (1872).
• G. Cantor, Uber Unendliche, Lineaere Punktmannigfaltigkeiten, Ibid., Vol.20, No.III, pp.113-121 (1882).
• S. Ahmad, Absolutely Independent Axioms for the Derived Set Operator, The American Mathematical Monthly, Vol.73, No.4, pp.390-392 (1966).
• 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).
• 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).
• D. Cenzer and D. Mauldin, On the Borel Class of the Derived Set Operator, Bull. Soc. Math. France, Vol.110, pp.357{380 (1982).
• 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).
• M. M. Fr_echet, Sur Quelques Points du Calcul Functionnel, Rend. Circ. Matem. Palermo, Vol.22, pp.1-72 (1906).
• Y. Gnanambal, On Generalized Preregular Closed Sets in Topological Spaces, Indian J. Pure Appl. Math., Vol.28, pp.351-360 (1997).
• F. Harary, A Very Independent Axiom System, Monthly, Vol.68, pp.159-162 (1961).
• F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly, Vol.70, No.10, pp. 1085-1086 (1963).
• 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)
• D. Higgs, Iterating the Derived Set Function, The American Mathematical Monthly, Vol.90, No.10, pp.693-697 (1983).
• H. J. Kowalsky, Topologische Raume, Birkhauser, Verlag, Basel und Stuttgart, pp.53 (1961).
• R. M. Latif, Characterizations and Applications of -Open Sets, Soochow Journal of Mathematics, Vol.32, No.3, pp.1-10 (2006).
• Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science, Vol.345, pp.63-76 (2019).
• 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).
• S. Modak, Some Points on Generalized Open Sets, Caspian Journal of Mathematical Sciences (CJMS), Vol.6, No. 2, pp.99-106 (2017).
• 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).
• A. Al-Omari and M. S. M. Noorani, On b-Closed Sets, Bull. Malays. Sci. Soc., Vol.32, No.1, pp.19-30 (2009).
• 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).
• 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).
• N. E. Rutt, On Derived Sets, National Mathematics Magazine, Vol. 18, No. 2, pp.53-63 (1943).
• 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).
• 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).
• R. Spira, Derived-Set Axioms for Topological Spaces, Portugaliar Mathematica, Vol.26, pp.165-167 (1967).
• C. Steinsvold, Topological Models of Belief Logics, Ph.D. Thesis, City University of New York (2007).
• J. Tucker, Concerning Consecutive Derived Sets, The American Mathematical Monthly, Vol.74, No.5, pp.555-556 (1967).
• J. Dixmier, General Topology, Springer Verlag New York Inc., Vol.18, pp.31-35 (1997).
• S. Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, Vol.18, pp.31-35 (1970).
• 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).
• 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).
• 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).
• 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).
• 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).
• 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).
• 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).
• M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R_eduit, Mauritius (2020).