Properties of 2-CNF mutually dual and self-dual T_0 -topologies on a finite set and calculation of T_0-topologies of a certain weight

Abstract

The problem of counting non-homeomorphic topologies as well as all topologies on an n-elements set is still open. The topologies with the weight k>2n-1, where k is the number of the elements of the topology on an n-elements set, which are called close to the discrete topology have been studied completely. Moreover R.~Stanley in 1971, M.~Kolli in 2007 and in 2014 have been found the number of T0-topologies on an n-elements set with weights k≥7·2n-4, k ≥3·2n-3, and k≥5·2n-4 respectively. In the present paper we investigate T0-topologies using the topology vector, being an ordered set of the nonnegative integers that define the minimal neighborhoods of the elements of the given finite set, and also using the special form of 2-CNF of Boolean function. In 2021 the authors found the form of the vector of T0-topologies with k≥5·2n-4 and the values k∈[5·2n-4, 2n-1], for which there are no T0-topologies with the weight k. The method of describing of T0-topologies using the special form of 2-CNF of Boolean function is used for the identification of the mutually dual and self-dual T0-topologies, and the properties of such 2-CNF Boolean function are used for counting T0-topologies with the weight 25·2n-6.