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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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/188421 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | 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| Summary: | 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).
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| ISSN: | 1726-3255 |