On Generalizations of Separation Properties, Compactness and Connectedness

Abstract

The thesis is concerned with generalizations of some important and interesting properties of separation, compactness and connectedness in a span of seven chapters. In the first chapter pseudo regular and pseudo normal topological spaces have been defined. Their properties have been studied and a number of important theorems regarding these spaces have been established. In the second chapter strongly pseudo-regular and strongly pseudo-normal topological spaces have been introduced and their properties have been studied. A number of important theorems have been proved in this regard. This is the third chapter. In this chapter strictly pseudo-regular and strictly pseudo-normal topological spaces have been defined and their properties have been studied. In the former class a compact set can be separated from an external point by a continuous function, while in the latter, two disjoint compact sets can be separated by a continuous function. Many important properties have been proved. The fourth chapter introduces the notions of nearly regular topological spaces of the first kind and the second kind and studies their properties. A number of important theorems regarding these spaces have been established. In this fifth chapter two new generalizations of normal spaces have been defined and studied. The spaces in these classes have been termed nearly normal topological spaces of the first kind and the second kind respectively. This is the sixth chapter. In this chapter further new generalizations of normal spaces have been made. These have been called slightly normal spaces of the first kind, the second kind and the third kind respectively. A number of important properties of these spaces have been proved. In this chapter seven compactness has been generalized to pseudo-compactness and c-compactness, and a continuum i.e., a connected compact space has been generalized to pseudo-continuum. Several properties of these three classes of spaces have been studied.