The properties of 2-CNF of the mutually dual and self-dual T0-topologies on the finite set and the calculation of T0-topologies of a certain weight

Abstract

The problem of counting non-homeomorphic topologies as well as all topologies on an n-element set is still open. The topologies with the weight k>2^n-1, where k is the number of the elements of the topology on an n-element 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-element set with weights k≥7·2^n-4, k ≥3·2^n-3, and k≥5·2^n-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≥5·2^n-4 and the values k∈[5·2^n-4, 2^n-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 25·2^n-6.