On a Type of Generalized GS_β-Closed Set in Grill Topology

Abstract

This work intends to propose the concept of generalized -closed set-in grill topological space. Further, some of the properties of such generalized -closed set are evaluated and provided with suitable illustrations. Many theorems based on generalized -closed set have also been proved in this work. Furthermore, the ideas of -regular space and -normal space are also studied. Also, the relationship between -regular and - normal spaces is discussed. The properties of  -regular space and -normal space concepts have also been discussed.
Introduction: Alo, R. A., and Shapiro, H. L  defined the semi- normal spaces and considered some of their important inheritance. The authors further studied the compactness in grill topological spaces and studied some of their properties. Data granulation serves as a vital decision-making tool in numerous real-life applications. Mashhour, A. S and Abd, M. E  introduced certain new topological tools employing rough set approximations for the purpose of data granulation. The introduction of  -normality helps to separate closed subsets, continuous functions. It also extends the support to define the concepts of complete regularity and Tietze extension theorem. The -regular space studied in this article is regular, that is completely regular, which is a stronger condition.
Objectives: I.Arockiarani and A. Karthika introduced a new concept such as generalized -closed set in grill topological space. The above paper motivates me to do the research in generalized -closed set.  In this paper a new concept of  “generalized -closed set in grill topological space” are introduced.  In this article we are using stronger condition for proving some theorems.
Methods: The present task ambitioned at formation of the extension of topological system applying the idea of grill. The importance of the theory of grill between the topological spaces have been elaborated. A set is countable collection based on elements, without integrity of form. At the same time some kinds of algebraic actions are applied on this set, indefinitely the elements of this set are correlated into a whole, so ultimately it grows into a space. So ultimately in the whole paper we are applying the concept of grill topological space.
Results: In this article a new concept of generalized -closed set in a grill topological space are introduced. Using this definition we are proving many results based on operators on grill topological space. Furthermore, the implication such as “all closed set are generalized -closed sets” is proved and the converse may not be true are also proved with proper illustration. In addition the relationship between -normal space, -normal space, -regular space and -regular space are also studied. Further many equivalent conditions are also proved using -normal space and -regular space.
Conclusions: The work proposes and investigates the concept of generalized -closed sets within the framework of grill topological spaces. It explores the properties of these sets, provides illustrative examples, and proves theorems related to their behavior. Additionally, the study delves into the concepts of -regular space and -normal space, discussing their properties and implications. The research contributes to the understanding of these concepts and their applications in the context of grill topological spaces.