Abstract

Let 𝔼 be a set with n elements, and let τ(n, k) be the set of all labelled topologies on 𝔼, having k open sets, and T (n, k) = |τ(n, k)|. In this paper, we use a direct approach to compute T (n, k) for all n ≥ 4 and k ≥ 6 · 2^{n−4}.