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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| Datum: | 2019 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2019
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543430395559936 |
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| author | Kizmaz, M. Yasir |
| author_facet | Kizmaz, M. Yasir |
| author_sort | Kizmaz, M. Yasir |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2019-03-23T17:44:10Z |
| description | 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 \(T_0(p+n)\equiv T_0(n+1) \ (mod \ p)\). |
| first_indexed | 2025-12-02T15:42:55Z |
| format | Article |
| id | admjournalluguniveduua-article-437 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:42:55Z |
| publishDate | 2019 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-4372019-03-23T17:44:10Z On the number of topologies on a finite set Kizmaz, M. Yasir topology, finite sets, \(T_0\) topology 11B50; 11B05 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 \(T_0(p+n)\equiv T_0(n+1) \ (mod \ p)\). Lugansk National Taras Shevchenko University 2019-03-23 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437 Algebra and Discrete Mathematics; Vol 27, No 1 (2019) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437/pdf Copyright (c) 2019 Algebra and Discrete Mathematics |
| spellingShingle | topology finite sets \(T_0\) topology 11B50 11B05 Kizmaz, M. Yasir On the number of topologies on a finite set |
| title | 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_short | On the number of topologies on a finite set |
| title_sort | on the number of topologies on a finite set |
| topic | topology finite sets \(T_0\) topology 11B50 11B05 |
| topic_facet | topology finite sets \(T_0\) topology 11B50 11B05 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/437 |
| work_keys_str_mv | AT kizmazmyasir onthenumberoftopologiesonafiniteset |