Geodesic
Method of construction of topologies on any finite set
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2012), pp. 29-42

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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 $k$. 
@article{BASM_2012_2_a2,
     author = {V. I. Arnautov},
     title = {Method of construction of topologies on any finite set},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {29--42},
     publisher = {mathdoc},
     number = {2},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2012_2_a2/}
}
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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/