THEORY OF GENERALIZED SEPARATION AXIOMS IN GENERALIZED TOPOLOGICAL SPACES

Year 2022, Volume: 5 Issue: 1, 1 - 23, 01.03.2022

Mohammad Irshad Khodabocus Noor-ul-hacq Sookıa

https://doi.org/10.33773/jum.974278

Cited By: 5

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

Thanks

The authors would like to express their sincere thanks to Prof. Endre Makai, Jr. (Professor Emeritus of the Mathematical Institute of the Hungarian Academy of Sciences) for his valuable suggestions.

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