Types of convergence in generalized metric spaces

The aim of this master's thesis is to systematically deal with the concepts of convergence in generalized metric spaces and to contribute theoretically to this field. In this context, this master's thesis is planned to consist of five main chapters. Firstly, in the introduction part, some convergence concepts, such as statistical and ideal convergence, their usage areas, properties, and historical processes on some metric spaces are mentioned. In the second chapter, the historical development of the problems studied in the thesis and recent studies are analyzed in detail. In the third chapter, entitled Sequences, Convergence and Metric Spaces, the classically defined metric spaces and the basic concepts of convergence in these spaces, the definition of generalized metric spaces, the basic properties of these structures, the topologies they form and the differences they carry in the context of the concept of convergence, quaternion-valued generalized metric spaces, double sequences and the convergence properties of these sequences, and recent studies on all these topics are included. In addition, some basic theorems that we will use in our results are presented. The next section, called Research Findings and Discussion, is divided into two subsections. In the first subsection of the fourth section, some results on statistical convergence and ideal convergence in generalized metric spaces are given. In the other subsection of this chapter, quaternion-valued generalized metric spaces are introduced by choosing quaternions as the image set. Accordingly, a broader notion of metric is constructed by utilizing the structural properties of quaternions, and the relations of these structures with convergence theory are investigated. In this direction, some results on statistical convergence and ideal convergence for even sequences in quaternion-valued generalized metric spaces are examined in detail. In this context, with the structural richness provided by quaternions, the convergence behavior of double sequences is investigated in a more general framework. The aim of the study is not only to extend the results in the existing literature but also to develop a more holistic understanding of the concept of convergence in different mathematical structures by creating a new theoretical ground. The last chapter of this thesis is the Conclusion and Recommendations section. In this section, which is the fifth chapter, the main results given in the previous chapter are briefly discussed, and some suggestions and problems related to the continuation of these results are given.