Geodesic
Estimation of the number of one-point expansions of a~topology which is given on a~finite set
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2011), pp. 17-22.

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Let $X$ be a finite set and $\tau$ be a topology on $X$ which has precisely $m$ open sets. If $t (\tau)$ is the number of possible one-point expansions of the topology $\tau$ on $Y=X\bigcup\{y\}$, then $\frac{m\cdot(m+3)}2-1\ge t(\tau)\ge2\cdot m+\log_2m-1$ and $\frac{m\cdot(m+3)}2-1=t(\tau)$ if and only if $\tau$ is a chain (i.e. it is a linearly ordered set) and $t(\tau)=2\cdot m+\log_2m-1$ if and only if $\tau$ is an atomistic lattice. 
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V. I. Arnautov. Estimation of the number of one-point expansions of a~topology which is given on a~finite set. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2011), pp. 17-22. http://geodesic.mathdoc.fr/item/BASM_2011_2_a1/

[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”, Buletinul Academiei de Stiinte a Respublicii Moldova. Matematica, 2010, no. 3(64), 67–76 | MR | Zbl

[2] Skornyakov L. A., Elements of the theory of structures, Nauka, Moscow, 1982, 147 pp. (Russian) | MR | Zbl

[3] Birkgoff G., Theory of lattices, Nauka, Moscow, 1984, 567 pp. (Russian)