The study of the topological structure on a finite set involves solving the problems of counting and enumerating topologies. For this purpose, topologies are modeled by graphs, matrices, Boolean functions, ordered sets of non-negative integers, which determine the minimal neighborhoods of the elements of a given finite set - vectors of topologies.
The theory of topologies on finite sets has many applications. In graph theory, the task of enumerating topologies on finite sets is equivalent to the task of enumerating transitive directed graphs, in algebraic topology it is reduced to enumerating homotopy types of finite sets. In chemistry, the topological structure on finite sets is used for the analysis of molecular structures, in the theory of pattern recognition - for digital image processing based on finite sets of observations in order to obtain information about the content of the image based on the concept of proximity of points.
The question of the total number of non-homeomorphic topologies, as well as the number of all topologies on an n-element set, remains open. To-topologies play an exclusive role in solving problems of enumeration and counting of topologies. Provided that a topology has m open sets, it is said to belong to the m-class (or has weight m).
The use of the topology vector made it possible to investigate all To-topologies with weight m>2^(n-1), which are called close to discrete topology. The dissertation uses the representation of topology by a vector, as well as the relationship between topologies on n-element and (n-1)-element sets, which is described using the topology consistency relation. The vectors of all To-topologies with weight 2^(n-2)<m≤2^(n-1) on the n-element set, which are consistent with close to discrete topologies on the (n-1)-element set, are listed. Among them, To-topologies with weight 2^(n-2)<m<5⋅2^(n-4) were investigated for the first time. In the works of Stanley in 1971 and Kolli in 2007 and 2014, the number of To-topologies on the n-elements of the set with weights k≥7⋅2^(n-4), k≥3⋅2^(n-3) was found and k≥5⋅2^(n-4). Comparison of the results obtained in the dissertation with the above-mentioned results made it possible to list the classes of topologies in which all topologies are consistent with topologies close to discrete, as well as to prove the existence of classes of topologies with weight m∈[5⋅2^(n-4),13⋅ 2^(n-5)), which are not exhausted by To-topologies consistent with those close to discrete on the (n-1)-element set, and dual to them. The structure of the topologies on the n-element set, which are not consistent with the topologies close to the discrete one on the (n-1)-element set, was studied, and the vectors of such topologies were found.
When modeling topologies with Boolean functions, a conjunctive normal form of a special form is assigned to each To-topology. In the dissertation, a new concept was introduced - the maximal 2-KNF Boolean function, which defines the topology on a finite set. The existence of a bijection between a set of topologies on an n-element set and a set of maximal 2-KNF of n variables is proved. A method for recognizing mutually dual and self-dual To-topologies has been developed, which is important for counting the number of To-topologies with a given weight.
Chapter 1 is devoted to the analysis of publications related to the topic of the dissertation. It is noted that the problem of enumerating and counting topologies on a finite set arose as a problem of graph theory and that the first results of the research of this problem are related to the use of graphs. It is pointed out that the study of topologies on a finite set has peculiarities, since such a topological structure is discrete, and that the study of this structure is closely related to the theories of partially ordered sets, Boolean functions, and homotopy topology. In the same section, the special role of To-topologies for the problem of counting all topologies on an arbitrary finite set is noted.
Chapter 2 analyzes the published results related to the topic of the dissertation research and presents the results obtained in the dissertation. Special attention is paid to the description of methods and models for representing topologies, to the assessment of their advantages and disadvantages...
