## T0-metrikimsi Uzayların Simetrisizliğine Yaklaşım Teorileri

##### Özet

The aim of this thesis is to construct various original metric approach theories speci c
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 rst 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 rst 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 de ned symmetricantisymmetric
connectedness theories, which enable the approximation of the distances
of the points of the T0-quasi-metric space to each other, through the symmetricantisymmetric
paths established with the other points between these points, form the
basis of this thesis. In the third chapter of the thesis, rstly 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
iii
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
di erent conditions.
As another new approach to asymmetry, the topological approach is discussed in
the fth 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 rst two subsections, and many useful results
have been reached with the help of examples.
In the last part of the fth 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 ndings obtained in the
thesis and open questions that could be the subject of future study are presented.

##### Bağlantı

http://hdl.handle.net/11655/26119##### Koleksiyonlar

**Hacettepe Üniversitesi Kütüphaneleri**

**Açık Erişim Birimi**

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