On The Number Of Topologies On A Finite Set

We denote the number of distinct topologies which can be defined on a set $X$ with $n$ elements by $T(n)$. Similarly, $T_0(n)$ denotes the number of distinct $T_0$ topologies on the set $X$. In the present paper, we prove that for any prime $p$, $T(p^k)\equiv k+1 \ (mod \ p)$, and that for each natural number $n$ there exists a unique $k$ such that $T(p+n)\equiv k \ (mod \ p)$. We calculate $k$ for $n=0,1,2,3,4$. We give an alternative proof for a result of Z. I. Borevich to the effect that $T_0(p+n)\equiv T_0(n+1) \ (mod \ p)$.