Geodesic
Method of construction of topologies on any finite set
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2012), pp. 29-42 
V. I. Arnautov. Method of construction of topologies on any finite set. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2012), pp. 29-42. http://geodesic.mathdoc.fr/item/BASM_2012_2_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let a topology $\tau$ be defined on a finite set. We give the definition of quasiatoms in the lattice $(\tau,\subseteq)$ and study their properties. For any splitting of a finite set $X$ into $k$ subsets we give a method of constructing any topology on the set $X$ for which this splitting is the set of all quasiatoms and the weight of this topological space is equal to $k$.

[1] Arnautov V. I., Kochina A. V., “The method of construction of one-point expansions of a topology on a finite set and its application”, Bul. Acad. Ştiinţe Repub. Moldova. Mat., 2010, no. 3(64), 67–76 | MR | Zbl

[2] Arnautov V. I., “Estimation of the number of one-point expansions of a topology which is given on a finite set”, Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2011, no. 2(65), 17–22 | MR | Zbl

[3] Skornyakov L. A., Elements of the theory of structures, Nauka, Moscow, 1982 (in Russian) | MR | Zbl

[4] Birkgoff G., Theory of lattices, Nauka, Moscow, 1984 (in Russian)