The Number of Topologies on a Finite Set

Abstract

Let X be a finite set having n elements. How many different labeled topologies one can define on X? Let T (n, k) be the number of topologies having k open sets. We compute T (n, k) for 2 ≤ k ≤ 12, as well as other results concerning T0 topologies on X having n + 4 ≤ k ≤ n + 6 open sets.