Geodesic
The number of topologies on a finite set
Journal of integer sequences, Tome 9 (2006) no. 2
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 $T_{0}$ topologies on $X$ having $n+4 = k = n+6$ open sets.
Classification : 11B73, 11B65
Keywords: labeled topology, Stirling number 
@article{JIS_2006__9_2_a6,
     author = {Benoumhani,  Moussa},
     title = {The number of topologies on a finite set},
     journal = {Journal of integer sequences},
     year = {2006},
     volume = {9},
     number = {2},
     zbl = {1103.11007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2006__9_2_a6/}
}
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Benoumhani,  Moussa. The number of topologies on a finite set. Journal of integer sequences, Tome 9 (2006) no. 2. http://geodesic.mathdoc.fr/item/JIS_2006__9_2_a6/