T0 and T1 reflexive relation spaces

T_0 and T_1 separation axioms are fundamental concepts in topology that define the separability of points within a topological space. The T_0 axiom states that in a topological space, any two distinct points must be distinguishable by at least one open set. On the other hand, the T_1 axiom requires that each point has open sets containing itself while excluding all other points. In T_1 spaces, each point has its own distinct open set, whereas T_0 spaces do not necessarily guarantee this separation. T_0 spaces are utilized in various fields, including automata theory, data science, machine learning, electronics, and digital logic design. This thesis comprises six chapters. The second chapter provides definitions of categorical concepts relevant to the study. The definitions corresponding to T_0 and T_1 topological spaces within topological categories are illustrated through examples, and their applications are discussed. In the third chapter, reflexive related spaces are introduced, and the RRel category is defined, consisting of objects as reflexive related spaces and morphisms as relation preserving functions. The properties of the RRel category are examined, demonstrating that it constitutes a normalized topological category. The fourth chapter establishes T_0 and T_1 objects within the RRel category, both in local and general contexts. At the end of this chapter, relationships between T_0 and T_1 objects in other topological categories are provided. In the fifth chapter, sober topological spaces are first introduced. Subsequently, sober, quasi-sober, and irreducible objects within the RRel category are characterized. Finally, concepts related to these objects in other categories and their interrelations are examined.