THEORY OF GENERALIZED SEPARATION AXIOMS IN GENERALIZED TOPOLOGICAL SPACES

Authors : Mohammad Irshad Khodabocus, Noor-ul-hacq Sookia

Pages : 1-23

Doi:10.33773/jum.974278

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Publication Date : 2022-03-01

Article Type : Research

Abstract :In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms is defined in terms of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) and the properties and characterizations of a Tg-space endowed with each such g-Tg,K, g-Tg,F, g-Tg,H, g-Tg,R, g-Tg,N-axioms are discussed. The study shows that g-Tg,F-axiom implies g-Tg,K-axiom, g-Tg,H-axiom implies g-Tg,F-axiom, g-Tg,R-axiom implies g-Tg,H-axiom, and g-Tg,N-axiom implies g-Tg,R-axiom. Considering the Tg,K, Tg,F, Tg,H, Tg,R, Tg,N-axioms as their analogues but defined in terms of corresponding elements belonging to the class of open, closed, semi-open, semi-closed, preopen, preclosed, semi-preopen, and semi-preclosed sets, the study also shows that the statement Tg,α-axiom implies g-Tg,α-axiom holds for each α ∈ {K, F, H, R, N}. Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.
Keywords : Generalized Topology, Generalized Topological Space, Generalized Separation Axioms, Generalized Sets