Summary: 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.

11B73, 11B65

labeled topology, Stirling number