Some relationships between quasi-uniform space and probabilistic metric space

In this compilation thesis, probabilistic metric spaces and quasi-uniform concepts are dealt with. First, the basic concepts of topological spaces, spaces of distribution functions, t-norm and triangle function concepts are given. Quasi-uniform spaces and their properties are introduced. Various examples are given to show that not every quasi-uniform is a regularity. By giving the proof that every quasi-uniform gives rise to a topology, some topological properties of this topology are examined. After, probabilistic metric space defined with the help of triangle function and Menger probabilistic metric space defined with the help of t-norm are given. For the probabilistic metric space concept defined with the help of the triangle function, it is mentioned that, under the appropriate continuity condition, a uniformity can be determined, so that the topology caused by the uniformity can be defined on this space. Similarly, for every point (p,q) ∈ X × X, the F(p,q) functions defined from the extended real numbers to the closed interval [0,1] are left-continuous, distribution functions defined with the help of the t-norm concept (X,F,t) given the Menger probabilistic metric space, it is stated that the set X is metrizable. Therefore, it has been observed that a uniformity is obtained from a metric topology on X and that the topology created by this uniformity coincides with the metric topology. In the last section, unlike the Menger probabilistic metric space, when the F(p,q) functions are not continuous from the left, given the (X,F,t) space defined with the help of the t-norm concept under certain conditions, the X set is a topological space with a topology caused by a quasi-uniformity. and some properties of this space are investigated. The relation of this topology with the metric topology on the Menger probabilistic metric space is given, it is obtained that it is first countable T2 (Hausdorff) and it is determined that this space is quasi-metric.