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

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.