Study of Some Properties of Topological Spaces and of Their Generalisations

Abstract

The various topological concepts and properties like openness, closedness, continuity, compactness, connectedness etc. have been generalised by mathematicians. In this thesis we have studied these generalisations and made our own contribution to define new concepts and properties and study them. In the first chapter a number of useful terms of set theory and topology and ring theory have been defined and a few important results have been stated without proof. Generalisations of different topological terms introduced previously by others have been described. These include generalisations of open sets like semi-open, pre-open, locally closed and a-sets, and generalisations of continuity like semi-continuity, somewhat continuity, somewhat semi-continuity, D •-continuity. The second chapter is a collection of useful topological results which have been proved for application in later chapters. In the third chapter a numbers of generalisations of open and closed sets have been defined and their distinctness has been established. These as well as earlier known such generalisations have been compared and relationship of inclusion among themselves established. Investigation has been made as which ones among these form topologies. The inclusion relations among the topologies generated by these classes have been studied. The fourth chapter is an exposition of a few special topologies, Topological spaces and their properties. Although most of these lack the kind of qualities one of would like them to have, some of them occur naturally in a number of important branches of Mathematics. The fifth and the last chapter treats some Generalisations of connectedness and compactness. 0-connected and H-closed spaces and H-continua have been studied.