Operation Approach of gs Open Sets in Topological Spaces

Abstract

Topology, as so many other branch of mathematics, evolved out of the revolutionary changes undergone by the concept of geometry during the 19th century. Topologists freed themselves into a narrow approach to work in abstract spaces such as n-dimensional manifolds, projective spaces, Riemann surfaces, function space etc. Topology has developed as a field of study from geometry and set theory, through analysis of concepts such as space, dimension and transformation and emerged with the rudimentary knowledge of Set Theory, Algebra, Geometry and Analysis. The sole purpose of the topology is to elucidate and investigate the ideas of continuity and connectivity within the frame work of mathematics. The study of topological spaces and general properties makes up one branch of topology known as general topologyand#8223;. General Topology has exhibited its elegance in both pure and applied aspects so that one can visualize the existence of topological structures and the usage of topological properties in various application fields. newlineIn 1979, Kasahara[29] introduced the concepts of operation in topological spaces and operation closed graph of a function. Several known characterizations of compact spaces, H-closed spaces and nearly compact spaces are unified by generalizing the notion of compactness with the help of a certain operation of a topology into the power set , by choosing some special mappings such as as the identity mapping, the closure operation or the interior closure operation. In 1983, Jankovic[19] introduced and studied the concept of operation-closures of a subset, operation closed sets in a topological space and several related topics using the concept of -closed sets and the -closed graphs. H. Ogata[18] introduced the notion of operation open sets in topological spaces and operation-separation axioms of topological spaces. Several forms of operation approaches have been introduced and analyzed by many topologists. Motivated by the study of operation approaches, the author has defined and studied a new concept na