Generalizations of Some Properties of Topological and Bitopological Spaces

Abstract

The thesis is concerned with generalizations of some important and interesting properties of topological and bitopological spaces in a span of four chapters. The first chapter constitutes an introduction and study of (i) a weak form of strong continuity, (ii) RC- continuity, (iii) perfect continuity, (iv) contra- precontinuity and (v) contra continuity in bitopological spaces . It thus generalizes the corresponding concepts in topology introduced by Donchev, Jafari and Noiri and studied by them. In addition, generalizing the works of Ekici and Noiri, as investigation of relationships between graphs and contra δ-precontinuous functions in bitopological spaces has also been made in this chapter. In the second chapter the problems of δ-compactness of topological spaces has been generalized to the corresponding properties in bitopological spaces. Some important properties of δ-compactness in bitopological spaces have been established, which are generalizations of results of park, Srivastava and Gupta. Also, a characterization of δ-Hausdorff bitopological spaces has been made and some properties of such spaces have been established, generalizing results of Srivastava and Gupta. The third chapter introduces the notions of weakly β-continuous functions in tritopological spaces and investigates several properties of these functions, thus generalizing the corresponding works in topological spaces by Khedr, Al-Areefi and Noiri and in bitopological spaces by Tahiliani. In the fourth chapter the idea of density topology has been introduced for tritopological spaces and has been used to prove certain theorem involving some separation properties. The concept of density of sets in a tritopological spaces and the notion of its trioclosure generalizing topology have been introduced and fruitfully used for study of separation properties.