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

Year 2023, Volume: 5 Issue: 1, 17 - 36, 18.07.2023

Mohammad Irshad Khodabocus , Noor-ul-hacq Sookıa

https://doi.org/10.47086/pims.1214055

Cited By: 1

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.

Keywords

Generalized topological space, generalized sets, generalized interior operator, generalized interior operator, generalized closure operator

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