Generalized Topological Operator Theory in Generalized Topological Spaces: Part II. Generalized Interior and Generalized Closure

Abstract

In a recent paper (Cf. [19]), we have presented the definitions and the essential properties of the generalized topological operators g-Int_g, g-Cl_g : P (Ω) −→ P (Ω) (g-T_g-interior and g-T_g-closure operators) in a generalized topological space T_g = (Ω, T_g) (T_g-space). Principally, we have shown that (g-Int_g , g-Cl_g) : P (Ω) × P (Ω) −→ P (Ω) × P (Ω) is (Ω, ∅)-grounded, (expansive, non-expansive), (idempotent, idempotent) and (∩, ∪)-additive. We have also shown that g-Int_g : P (Ω) −→ P (Ω) is finer (or, larger, stronger) than int_g : P (Ω) −→ P (Ω) and g-Cl_g : P (Ω) −→ P (Ω) is coarser (or, smaller, weaker) than cl_g : P (Ω) −→ P (Ω). In this paper, we study the commutativity of g-Int_g , g-Cl_g : P (Ω) −→ P (Ω) and T_g-sets having some (g-Int_g, g-Cl_g)-based properties (g-P_g, g-Q_g-properties) in T_g-spaces. The main results of the study are: The g-T_g-operators g-Int_g, g-Cl_g : P (Ω) −→ P (Ω) are duals and g-P_g-property is preserved under their g-T_g-operations. A T_g-set having g-P_g-property is equivalent to the T_g-set or its complement having g-Q_g-property. The g-Q_g-property is preserved under the set-theoretic ∪-operation and g-P_g-property is preserved under the set-theoretic {∪, ∩, C}-operations. Finally, a T_g-set having {g-P_g , g-Q_g}-property also has {P_g , Q_g }-property.