Theories of approximation to the asymmetry of t0-quasi-metric spaces

The aim of this thesis is to construct various original metric approach theories specific to the asymmetric environment for the asymmetry of T0-quasi-metrics, non-metrics and also known as asymmetric distance functions, that is, to determine how close or far T0-quasi-metrics are from being a metric. In the first chapter of the thesis, which consists of six chapters, the main ideas on which it is based are mentioned and an introduction to the subject of the thesis is made. Some of the basic features of T0-quasi-metrics and various asymmetric structures developed in this environment are reminded in the first part of the second chapter, after that the new results obtained from these structures are presented in the second part. The last part of this chapter is devoted to various examples of T0-quasi-metric spaces, are studied in detail which we will use throughout the thesis. Considering the symmetry feature of the metric, the previously defined symmetric-antisymmetric connectedness theories, which enable the approximation of the distances of the points of the T0-quasi-metric space to each other, through the symmetric-antisymmetric paths established with the other points between these points, form the basis of this thesis. In the third chapter of the thesis, firstly the details of these theories are reminded, and in the second part, new results and examples that we have obtained within the framework of the relevant theories are mentioned. In the fourth chapter, as another original work; the theories of symmetric and antisymmetric connection extensions are established for a T0-quasi-metric space. In particular, it is proved that every bounded T0-quasi-metric space has a symmetrically connected one-point extension, and every metric space has an antisymmetrically connected one-point extension. Also, ``Does every T0-quasi- metric space have an antisymmetrically connected extension?" question is investigated, and the positive answers are given to this question by means of (counter)examples as well as theorems involving different conditions. As another new approach to asymmetry, the topological approach is discussed in the fifth chapter. In this framework, local symmetric and local antisymmetric connectedness theories, which are natural localizations of symmetric and antisymmetric connectedness theories according to the symmetrization topology of T0-quasi-metric, are constructed. All the properties of locally (anti)symmetrically connected spaces such as their relations with other structures, their inheritance in subspaces, products, etc. have been investigated in detail in the first two subsections, and many useful results have been reached with the help of examples. In the last part of the fifth chapter, asymmetric norm theory, which is a milestone in their development in asymmetric topology by producing T0-quasi-metrics, is considered as another alternative working environment in order to approach to the asymmetry of T0-quasi-metrics. The thesis is completed with the last chapter, in which the findings obtained in the thesis and open questions that could be the subject of future study are presented.