Geodesic
On the number of topologies on a finite set
Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 50-57 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{ADM_2019_27_1_a5,
     author = {M. Yasir Kizmaz},
     title = {On the number of topologies on a finite set},
     journal = {Algebra and discrete mathematics},
     pages = {50--57},
     year = {2019},
     volume = {27},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a5/}
}
TY  - JOUR
AU  - M. Yasir Kizmaz
TI  - On the number of topologies on a finite set
JO  - Algebra and discrete mathematics
PY  - 2019
SP  - 50
EP  - 57
VL  - 27
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a5/
LA  - en
ID  - ADM_2019_27_1_a5
ER  - 
%0 Journal Article
%A M. Yasir Kizmaz
%T On the number of topologies on a finite set
%J Algebra and discrete mathematics
%D 2019
%P 50-57
%V 27
%N 1
%U http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a5/
%G en
%F ADM_2019_27_1_a5
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/

[1] N. J. A. Sloane, Online Encyclopedia of Integer Sequences, Concerned with sequences A000798, A001035

[2] N. J. A. Sloane, Online Encyclopedia of Integer Sequences, A265042

[3] M. Benoumhani, “The number of topologies on a finite set”, J. Integer Seq., 9:2 (2006), 06.2.6 | MR | Zbl

[4] Z. I. Borevich, “Periodicity Of Residues of The number Of Finite Labeled Topologies”, Journal of Soviet Mathematics, 24:4 (1984), 391–395 | DOI | MR | Zbl

[5] Z. I. Borevich, “Periodicity of residues of the number of finite labeled $T_0$-topologies”, Modules and algebraic groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 114, 1982, 32–36 (Russian) | MR | Zbl

[6] M. Erné, “Struktur- und Anzahlformeln für Topologien auf endlichen Mengen”, Manuscr. Math., 11:3 (1974), 221–259 | DOI | MR | Zbl

[7] A. S. Mashhour, M. E. Abd El-Monsef and A. S. Farrag, “On the number of topologies on a finite set”, Delta J. Sci., 10:1 (1986), 41–65 | MR

[8] R. Stanley, “On the number of open sets of finite topologies”, J. Combinatorial Theory, 10 (197), 74–79 | DOI | MR | Zbl