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

Authors : Mohammad Irshad KHODABOCUS, Noorulhacq SOOKIA, Radhakhrishna Dinesh SOMANAH

Pages : 183-220

Doi:10.33773/jum.1295736

View : 28 | Download : 36

Publication Date : 2023-07-31

Article Type : Research Paper

Abstract :In a T_g-space T_g = insert ignore into journalissuearticles values(Ω, T_g);, the g-topology T_g : P insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(Ω); in T_g-spaces. We show that insert ignore into journalissuearticles values(g-Der_g, g-Cod_g); : P insert ignore into journalissuearticles values(Ω); × P insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(Ω); × P insert ignore into journalissuearticles values(Ω); is a pair of both dual and monotone g-T_g-operators that is insert ignore into journalissuearticles values(∅, Ω);, insert ignore into journalissuearticles values(∪, ∩);-preserving, and insert ignore into journalissuearticles values(⊆, ⊇);-preserving relative to g-T_g-insert ignore into journalissuearticles values(open, closed); sets. We also show that insert ignore into journalissuearticles values(g-Der_g, g-Cod_g); : P insert ignore into journalissuearticles values(Ω); × P insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(Ω); × P insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(Ω); → P insert ignore into journalissuearticles values(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 ope