Nykorovych S. Approximations in pseudometric spaces

The object of study is subsets with given properties of the pseudometric space. The methods of functional analysis and the theory of continuous domains were used in the course of the dissertation. The main scientific results presented for defence are new. They are presented in the thesis for the first time: The ordinal properties of the set P s(X) of all pseudometrics on a fixed set X and the set P sU(X) of all pseudo-ultrametrics on a fixed set X are described. It is proved that the partially ordered sets P s(X) and P sU(X) have no nontrivial approximation relations, hence they are not continuous or dual continuous, and the set P s(X) is not even a lattice. We prove that the partially ordered set CP sU(X) (the set of all compact pseudo-ultrametrics on a fixed set X) is continuous. It is shown that the approximation of pseudo-ultimetrics (in the sense of order theory) is effective only in the case of compact pseudo-ultimetrics. It is established which classes of pseudometrics on fixed sets are continuous and dual continuous. A method of constructing compact pseudo-ultrametrics significantly lower or significantly higher than the given one and arbitrarily close to it is obtained. For a fixed compact pseudo-ultimetric d on a set X, we propose methods for metrizing the set d of all compact pseudo-ultimetrics on X less than or equal to d, and prove that they are topologically equivalent (except for the Hartog de Vink metric). It is established that compact pseudo-ultrametrics continuous with respect to a given compact pseudo-ultrametric on a fixed set form a complete normalised idempotent vector space, generate the Banach space of symmetric continuous functions of two variables equal to zero on the diagonal, but in general do not generate a positive cone in this space.