APPLICATION OF DIFFERENT MODELS FOR RESEARCH TOPOLOGY ON FINITE SETS

The study of topological structure on a finite set includes solving problems of counting and enumerating topologies. For this purpose, topologies are modeled by graphs, matrices, Boolean functions, ordered sets of non-negative integers – topology vectors. The results of studies of topologies on a finite set are closely related to digital image processing based on finite sets of observations, that is, an attempt to understand the content of an image based on the concept of proximity of points. This paper provides a brief overview of methods for studying topologies on an n -element set. T0 -topologies play an exceptional role in solving the problems of enumerating and calculating the number of all topologies. It is convenient to say that when a topology has m open sets, it belongs to the m -class (or has weight m ). The use of topology vector made it possible to investigate all T0 -topologies with weight m 2n 1 (close to discrete topology), to describe all T0 -topologies on the n????-???? element set with weight 2n 1 2n m , which are compatible with topologies close to discrete topology on an n 1 -element set. Comparison of the obtained results with the results of the works Stanley R.P. 1971, Kolli M. 2007 2014 helped to list the classes of topologies in which all topologies are compatible with topologies close to discrete topology on an n 1 ????-???? element set, as well as to show that there are the classes of topologies with the weight m n n 5 2 4 ,13 2 5 , which are not exhausted by T0 -topologies compatible with topologies close to discrete topology on an n 1 -element set or dual to them. When modeling topologies using Boolean functions for each T0 -topology there is a single conjunctive normal form of a certain type (maximal 2-CNF). The use of 2-CNF of Boolean functions made it possible to develop a technique for recognizing mutually dual and self-dual T0 -topologies and counting the number of T0 -topologies with a given weight. This article studies T0 -topologies on the n -element set with weight 2n 1 2n m , which are not compatible with topologies close to discrete topology on an n 1 -element set. Topology vector as a model is used to study of these T0 -topologies.