Geodesic
Direct and elementary approach to enumerate topologies on a finite set
Journal of integer sequences, Tome 10 (2007) no. 3.

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Summary: 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}$.
Classification : 05A15, 06A07, 06A99
Keywords: binary relation, enumeration, finite set, finite topology, partial order, posets 
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     author = {Kolli, Messaoud},
     title = {Direct and elementary approach to enumerate topologies on a finite set},
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Kolli, Messaoud. Direct and elementary approach to enumerate topologies on a finite set. Journal of integer sequences, Tome 10 (2007) no. 3. http://geodesic.mathdoc.fr/item/JIS_2007__10_3_a6/