Bashova N. The classification and the calculation of the number of the topologies on the finite sets

The dissertation deals with the classification of the topologies on the finite sets and with the calculation of the topologies in some of the classes with the help of the finite directed graphs and Boolean functions. The correspondence between the set of the topologies on the -element set and the set of the directed graphs of the special kind ( -quivers) has been set up. The properties of the -quivers of the -topologies have been studied. The following apparatus has been developed for the prove the theorem, connected with the existence and the structure of the topologies in -classes when : the concepts of the index of the element of the topology, the vector of the topology and the depth of the set have been proposed; the method of the estimation of the number of the elements of the topology depending on its vector has been given. The correspondence between the topologies on the -element set and the Boolean functions of variables has been set up. It has been proved that the Boolean function, which defines the topology on the finite set, is bijunctive, 0-satisfiable and 1-satisfiable Boolean function with 2-conjunctive normal form of the certain kind.