On the number of topologies on a finite set

Abstract

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 \(T_0(p+n)\equiv T_0(n+1) \ (mod \ p)\).