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

Abstract

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