On the number of topologies on a finite set
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...
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| Дата: | 2019 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2019
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| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| Резюме: | 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 \(T_0(p+n)\equiv T_0(n+1) \ (mod \ p)\). |
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