Geodesic
On the number of topologies on a finite set
Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 50-57.

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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. 
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M. Yasir Kizmaz. On the number of topologies on a finite set. Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 50-57. http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a5/

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