On the Cardinality of the T0-Topologies on a Finite Set

Abstract

Let T0(n,k) be the number of all labeled T0-topologies having k open sets that we can define on n points, and let t0(n,k) be the number of those which are nonhomeomorphic. In this paper, we compute these numbers for k≥5·2n-4 and arbitrary n≥4. The numbers tn0(n,k) of all unlabeled and non-T0-topologies with k open sets are also given for k≥2n-2.