On The Number Of Topologies On A Finite Set

TL;DR

This paper investigates the number of topologies on finite sets, providing modular arithmetic properties for these counts when the set size is a prime power or related to prime numbers, and offers new proofs for existing results.

Contribution

It establishes new congruence relations for the counts of topologies and $T_0$ topologies on finite sets, including prime power cases and related structures.

Findings

Proves that T(p^k) ≡ k+1 (mod p) for prime p.

Shows a unique k exists such that T(p+n) ≡ k (mod p) for each n.

Provides an alternative proof for Borevich's result on T_0 topologies.

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