On the number of topologies on a finite set

• Date: 2019
• Main Author: Kizmaz, M. Yasir
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2019
• Subjects: topology; finite sets; \(T_0\) topology; 11B50; 11B05
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437
• Journal Title: Algebra and Discrete Mathematics

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 th...