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, 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(pᵏ) ≡ k + 1 (mod p), and that for each natural number n there exists...
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| Дата: | 2019 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2019
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/188421 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the number of topologies on a finite set / M.Y. Kizmaz // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 50–57. — Бібліогр.: 8 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1884212023-03-01T01:27:00Z On the number of topologies on a finite set Kizmaz, M.Y. 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(pᵏ) ≡ 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). 2019 Article On the number of topologies on a finite set / M.Y. Kizmaz // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 50–57. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC: Primary 11B50, Secondary 11B05. http://dspace.nbuv.gov.ua/handle/123456789/188421 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
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(pᵏ) ≡ 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). |
| format |
Article |
| author |
Kizmaz, M.Y. |
| spellingShingle |
Kizmaz, M.Y. On the number of topologies on a finite set Algebra and Discrete Mathematics |
| author_facet |
Kizmaz, M.Y. |
| author_sort |
Kizmaz, M.Y. |
| title |
On the number of topologies on a finite set |
| title_short |
On the number of topologies on a finite set |
| title_full |
On the number of topologies on a finite set |
| title_fullStr |
On the number of topologies on a finite set |
| title_full_unstemmed |
On the number of topologies on a finite set |
| title_sort |
on the number of topologies on a finite set |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2019 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188421 |
| citation_txt |
On the number of topologies on a finite set / M.Y. Kizmaz // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 50–57. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT kizmazmy onthenumberoftopologiesonafiniteset |
| first_indexed |
2023-10-18T23:08:19Z |
| last_indexed |
2023-10-18T23:08:19Z |
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1796157347091972096 |