On The Number Of Topologies On A Finite Set

2015/03/28 by M. Yasi̇r Kızmaz, Kızmaz, Muhammet Yasir · 1 citation
Computer Science · Mathematics · #11B50 #Analytic Number Theory Research #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1503.08359

openalex publication_date 2015/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We denote the number of distinct topologies which can be defined on a set X with n elements by T(n). Similarly, T₀(n) denotes the number of distinct T₀ topologies on the set X. In the present paper, we prove that for any prime p, T(p^k)≡ k+1 (mod p), and that for each natural number n there exists a unique k such that T(p+n)≡ 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 T₀(p+n)≡ T₀(n+1) (mod p).

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