Version 1
: Received: 29 November 2020 / Approved: 30 November 2020 / Online: 30 November 2020 (20:21:55 CET)
How to cite:
KHODABOCUS, M.I.; SOOKIA, N. Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks. Preprints2020, 2020110747. https://doi.org/10.20944/preprints202011.0747.v1
KHODABOCUS, M.I.; SOOKIA, N. Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks. Preprints 2020, 2020110747. https://doi.org/10.20944/preprints202011.0747.v1
KHODABOCUS, M.I.; SOOKIA, N. Theory of g-Tg-Derived and g-Tg-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks. Preprints2020, 2020110747. https://doi.org/10.20944/preprints202011.0747.v1
APA Style
KHODABOCUS, M.I., & SOOKIA, N. (2020). Theory of g-T<sub>g</sub>-Derived and g-T<sub>g</sub>-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks. Preprints. https://doi.org/10.20944/preprints202011.0747.v1
Chicago/Turabian Style
KHODABOCUS, M.I. and Noor-Ul-Hacq SOOKIA. 2020 "Theory of g-T<sub>g</sub>-Derived and g-T<sub>g</sub>-Coderived Operators: Definitions, Essential Properties, Iterations, and Ranks" Preprints. https://doi.org/10.20944/preprints202011.0747.v1
Abstract
In a generalized topological space Tg = (Ω, Tg) (Tg-space), the g-topology Tg : P (Ω) −→ P (Ω) can be characterized in the generalized sense by specifying the generalized open, generalized closed sets (g-Tg-open, g-Tg-closed sets), generalized interior, generalized closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω) (g-Tg-interior, g-Tg-closure operators), or generalized derived, generalized coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω) (g-Tg-derived, g-Tg-coderived operators), respectively. For very many Tg-spaces, the δth-iterates g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω) of g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, defined by transfinite recursion on the class of successor ordinals are also themselves g-Tg-derived, g-Tg-coderived operators for new g-topologies in the generalized sense on Ω. Thus, the use of novel definitions of g-Tg-derived, g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, based on a very clever construction, together with their δth-iterates g-Tg-operators g-Derg(δ), g-Codg(δ) : P (Ω) −→ P (Ω), defined by transfinite recursion on the class of successor ordinals, will give rise to novel generalized g-topologies on Ω. The present authors have been actively engaged in the study of g-Tg-operators in Tg-spaces. The study of the essential properties and the commutativity of novel definitions of g-Tg-interior and g-Tg-closure operators g-Intg, g-Clg : P (Ω) −→ P (Ω), respectively, in Tg has formed the first part, and the study of the essential properties and sets of consistent, independent axioms of novel definitions of g-Tg-exterior and g-Tg-frontier operators g-Extg, g-Frg : P (Ω) −→ P (Ω), respectively, has formed the second part. In this work, which forms the last part on the theory of g-Tg-operators in Tg-spaces, the present authors propose to present novel definitions and the study of the essential properties of g-Tg-derived and g-Tg-coderived operators g-Derg, g-Codg : P (Ω) −→ P (Ω), respectively, and their δth-iterates, and the notions of g-Tg-open and g-Tg-closed sets of ranks δ in Tg-spaces.
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
