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

In a generalized topological space Tg = (Ω, ) (Tg -space), various ordinary operators -operators), namely, int_g, cl_g, ext_g, fr_g, der_g, 
 cod_g : P (Ω) −→ (T_g-interior, T_g-closure, T_g-exterior, T_g-frontier, T_g-derived, T_g-coderived operators), are defined in terms of sets (T_g-sets). Accordingly, T_g-operators (g-T_g-operators), g-Int_g, g-Cl_g, g-Ext_g, g-Fr_g, g-Der_g, g-Cod_g (g-T_g-interior, g-T_g-closure, g-T_g-exterior, g-T_g-frontier, g-T_g-derived, g-T_g-coderived operators) may be T_g-sets (g-T_g-sets), thereby making g-T_g-operators theory T_g-spaces an interesting subject inquiry. this paper, we present the definitions and essential properties g-T_g-interior g-T_g-closure g-Int_g , g-Cl_g (Ω), respectively, new class g-T_g-sets which studied earlier. The outstanding results to study has led are: Firstly, (g-Int_g, g-Cl_g) × is ∅)-grounded, (expansive, non-expansive), (idempotent, idempotent) (∩, ∪)-additive. Secondly, finer (or, larger, stronger than int_g coarser smaller, weaker) cl_g (Ω). elements supporting these facts reported therein as sources inspiration for more operations.