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

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