Abstract:

Let $\mathbb{E}$ be a set with $n$ elements, and let $\tau (n,k)$ be the set of all labelled topologies on $\mathbb{E}$, having $k$ open sets, and $T(n,k)=\left\vert \tau (n,k)\right\vert$. In this paper, we use a direct approach to compute $T(n,k)$ for all $n\geq 4$ and $k\geq 6\cdot 2^{n-4}$.