On The Number Of Topologies On A Finite Set

Muhammet Yasir Kızmaz

Published 2015-03-28Version 1

We denote the number of distinct topologies which can be defined on the 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 non-negative integer $n$ there exists a unique $k$ such that $T(p+n)\equiv k$. We calculate $k$ for $n=1,2,3,4$. We give an elementary proof for a result of Z.I.Borevich to the effect that $T_0(p+n)\equiv T_0(n+1) \ mod \ p$.