Analysis on properties of 𝝀𝒈𝜶-closed sets in topological spaces

Abstract

Generalizations to the closed sets in topological spaces was initially framed by Levine [1970], which formulated the approach to the study of generalizations with the respective closed sets. Fundamental theorems and properties which were given by him is of much importance in a unique manner that it is functional even today for the topologists in this field. This thesis is one such study which is dedicated to the analysis of -closed sets in topological spaces. The name -closed sets was given since we have implemented the concepts of -closure and -open sets. -open sets was defined by Njastad [1965] and -sets was primarily studied by Francisco G. Arenas et al. [1997]. They both have presented their respective concepts competently with all the distinct properties and theorems. newline-closed sets are the sets which broaden the area of research of -sets and -open sets. Initially, we have introduced the definition of -closed sets in topological spaces and studied the very usual basic properties of it. Diagrammatic representations of dependencies and independencies are also provided. Moreover the -closure and -interior operators are defined and examined. Following the introduction of -closed sets, we have proposed the continuity concepts such as -continuous maps and -irresolute maps and analyzed their properties. All the theorems and characterizations are derived sequentially. Various special forms of -continuous maps namely quasi -continuous, perfectly -continuous, totally -continuous and strongly -continuous maps are also defined and their interrelations are derived. Later, the idea of contra -continuous maps and contra -irresolute maps, their associations, properties and theorems are presented. Further, the notion of -homeomorphisms are introduced and analysed with the help of -closed maps and -open maps. Following this, -quotient maps have been defined and its properties are derived. Finally bi-contra -continuous maps and its special forms are defined with accurate examples.