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\begin{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 $k\geq 6\cdot 2^{n-4}$.
\end{abstract}

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