Seliutin D. Filters, generalized types of convergence and its applications

The thesis is devoted to the study of filters, their properties, and their applications to various branches of mathematics, in particular, the theory of topological vector spaces, general topology, and the theory of integration. The first chapter of the dissertation is devoted to the history of the study of the issues discussed in the dissertation and the literature review. The main constructions, which will be discussed throughout the dissertation, are described. The main known results, which will be used in the following sections, are also formulated. The second chapter of the dissertation is devoted to the study of the properties of filters and ultrafilters on a set of natural numbers. In the third chapter of the dissertation, we study ideals on the set of natural numbers N. The fourth chapter of the dissertation is devoted to the study of one, but rather wide class of ideals, namely, ideals generated by modular functions. In the fifth chapter of the dissertation, we study the application of the theory of filters to the theory of topological vector spaces. The completeness of a space is one of the most important concepts not only in the theory of topological vector spaces, but also in mathematics in general. That is why it is natural to generalize this concept for spaces that do not have a metric structure, in which we cannot use the language of convergence of sequences in the standard sense of the word. Therefore, in this section, we introduce a number of generalizations of the concept of completeness of space in the language of filters, analyze the connection between the introduced concepts and previously known ones, and investigate the properties of new types of completeness of space. The sixth chapter of the dissertation is devoted to the application of the theory of filters to the theory of integration.