Some topological properties on frames

The aim of this thesis is to explore the concept of frame, which is a special kind of a lattice, and to examine their topological properties, such as separation axioms and compactness. The study consists of six chapters. The first chapter provides a brief introduction to the thesis. In the second chapter, partially ordered sets, the basis of lattice theory, are discussed, as well as some special structures based on them. In addition, the concept of lattice is introduced and some basic definitions and concepts related to category theory are given. The third chapter examines the concepts of frame and locale and some structures derived from these concepts. Furthermore, the concept of sublocales and the structures of localic mappings are discussed. In the fourth chapter, the concepts of subfitness and fitness, which do not exist in the theory of topological spaces, are introduced. Moreover, local theory versions of the Hausdorff axiom are presented. This section also analyses the relations between all these axioms. The fifth chapter introduces the concept of compactness and explores its relations with separation axioms. Additionally, it presents a point-free counterpart to compactification. The final chapter introduces the valuation map and the metric derived from this map. Key Words: Lattice, Frame, Locale, Sublocale, Localic Mapping, Compact, Partially Ordered Set, Category Theory, Subfit, Fit, Hausdorff, Valuation.