New Characterizations of Topological Spaces

Abstract

In this paper, we introduce and investigate topological spaces called sgw-compactness Spaces and sgw-connectedness space and we get several characterizations and some of their properties. Also we investigate its relationship with other types of functions. Mathematics Subject Classification: 54D05, 54D30.

Introduction

I. INTRODUCTION

The notions of compactness and connectedness are useful and fundamental notions of not only general topology but also of other advanced branches of mathematics. Many researchers have investigated the basic properties of compactness and connectedness. The productivity and fruitfulness of these notions of compactness and connectedness motivated mathematicians to generalize these notions. In the course of these attempts many stronger and weaker forms of   compactness and connectedness have been introduced and investigated. D. Andrijevic [1] introduced a new class of generalized open sets in a topological space called b-open sets. The class of  b-open sets generates the same topology as the class of  b-open sets. Since the advent of this notion, several research paper with interesting results in different respects came into existence . M. Ganster and M.Steiner [5] introduced and studied the properties of gb-closed sets in topological spaces. The aim of this paper is to introduce the concept of sgw-compactness and sgw-connectedness in topological spaces and is to give some characterizations of sgw-compact spaces in terms of nets and filter bases. S.S. Benchalli, T.D. Rayanagoudar and P.G. Patil introduced the concept of g*-closed sets and S.S. Benchalli, T.D. Rayanagoudar and P.G. Patil and Shik John studied the concept of g*- preregular, g*- pre normal and obtained their properties by utilizing g*-closed sets. The notation of closed set is fundamental in the study of topological spaces. In 1970, Levine  introduced the concept of generalized closed sets in the topological space by comparing the closure of subset with its open supersets. The  investigation  on   generalization of   closed   set has lead to significant contribution   to  the   theory of separation axiom, covering properties   and   generalization of continuity. T. Kong, R. Kopperman and P. Meyer  shown some of the properties of generalized closed set have been found to be useful in computer science and digital topology. Caw, Ganster and Reilly  and   has shown that generalization of closed set is also useful to characterize certain classes of topological spaces and there variations, for example the class of extremely disconnected spaces and the class of submaximal spaces. In 1990, S.P. Arya  and T.M. Nour define generalized semi-open sets, generalized semi closed sets and use them to obtain some cauterization of s-normal spaces.

In 1993,N. PalaniInappan and K. Chandrasekhara Rao  introduced regular generalized closed (briefly rg-closed) sets and study there properties relative to union, intersection and subspaces. In 2000, A. Pushpalatha  introduce new class  of closed set called weakly closed (briefly w-closed) sets and study there properties.In 2007, S.S. Benchalli and R.S. Wali  introduced the new class of the set called regular w-closed (briefly rw-closed) sets in topological spaces. In this this paper is to introduce and study two new classes of spaces,namely Semi weakly generalized-normal and Semi weakly generalized- regular spaces and obtained their properties by utilizing Semi weakly generalized-closed sets.

References

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Copyright © 2023 Dr. Arun Kumar Gali. 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.