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Algebra and Discrete Mathematics, 2019, Volume 27, Issue 1, Pages 50–57
(Mi adm691)
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RESEARCH ARTICLE
On the number of topologies on a finite set
M. Yasir Kizmaz Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey
Abstract:
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 $p$, $T(p^k)\equiv k+1 \pmod p$, and that for each natural number $n$ there exists a unique $k$ such that $T(p+n)\equiv k \pmod p$. 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 $T_0(p+n)\equiv T_0(n+1) \pmod p$.
Keywords:
topology, finite sets, $T_0$ topology.
Received: 31.03.2017 Revised: 06.10.2017
Citation:
M. Yasir Kizmaz, “On the number of topologies on a finite set”, Algebra Discrete Math., 27:1 (2019), 50–57
Linking options:
https://www.mathnet.ru/eng/adm691 https://www.mathnet.ru/eng/adm/v27/i1/p50
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