A New Class of Open Sets using δ-Preopen Sets

Abstract

The scope of this thesis is to introduce a new class of sets called and#948;PS-open sets in topological spaces. Combining the concepts of and#948;-preopen and PS-open sets, a new class of sets called and#948;PS-open sets is introduced in this article. This class of sets lies between the classes of PS-open and and#948;-preopen sets. The behaviour of and#948;PS-open sets in various spaces such as locally indiscrete, hyperconnected, extremally disconnected, semi-T1, s-regular spaces are discussed and various interesting results are obtained. Further, we discuss some vital relations and interesting characterizations of -neighborhood. In a consequent manner, concepts of -Frontier, -Boundary, -Exterior and -Saturated Set are studied. Grill is a powerful tool in dealing with proximity spaces, closure spaces and in the theory of compactification. It is efficient in dealing with many topological situations. With this motivation, -open sets are analyzed through grills. newlineSeparation axioms are one among the most common, important and interesting concepts, four new types of spaces are defined and its properties are analyzed. Also, -convergence, -accumulation, -open cover are defined and using these results -Compactness is defined. Some of the existing results were analysed using the newly defined spaces. newlineAlso, various types of continuities using -open sets namely -continuity, quasi -continuity, perfectly -continuity, totally -continuity, strongly -continuity and contra -continuity are defined and their properties are discussed. Some weaker form of -continuity are defined and analysed. Further, we introduced somewhat -continuous, somewhat almost -continuous, somewhat -irresolute, somewhat -open and somewhat almost -open functions. These findings result in procuring several characterizations, properties and interrelations with other types of functions. newlineThe notions of -open and -closed functions are taken for study and their behaviours are characterized in locally indiscrete space. Further -irresoluteness and contra -irresoluteness are defined and properties are a