Some topological properties generalized to diframes

The aim of this thesis is to define the notion of diframe as a generalization of ditopological texture spaces and to study the topological concepts such as separation axioms and compactness in diframe setting. This work consists of six chapters. In the first chapter, we give a brief introduction. In the second chapter, we present some necessary preliminaries including frame theory and ditopological texture spaces. Chapter three is devoted to the study of coframes. We provide some new definitions and properties dual to those in frame theory. In chapter four, we establish the category of diframes. We first provide a link between morphisms of the category drTex of texture spaces and the category frames (Frm) and then obtain a full subcategory frTex of Frm. This connection allows us to construct the category diFrm of diframes and diframe homomorphisms. In this chapher, we also give the definitions of base, subbase and subdilocale of a diframe. In chapter five, we study separation axioms in diframes. In particular, we provide alternative characterizations of these axioms and investigate the connections between them. The final chapter deals with the compactness and stability in diframes. This chapter is divided into two section. In the first section, we discuss the questions of whether these properties are hereditary, and whether they are preserved by any reasonable kind of homomorphisms. Since stability is a property relating the frame and the coframe parts of a diframe, we replace compactness by stability to obtain diframe versions of topological results relating separation axioms and compactness. We also give a generalization of Alexander's subbase theorem. In the second section, we introduce two main concepts, that of locally compactness and locally stability in diframes. These concepts are defined in terms of suitable binary relations whereas their bitopological versions use the notion of neighbourhood which is a point-based structure. We also show that locally compactness and locally stability are preserved by morphism satisfying appropriate conditions.