Geodesic
Method for constructing one-point expansions of a~topology on a~finite set and its applications
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2010), pp. 67-76.

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The article consists of two parts. In the first part we present an algorithm which allows to receive, for any topology τ which is given on a set X from n elements, all topologies on the set X⋃{y} each of which induces the topology τ on the set X. In the second part (as an example) this algorithm is applied for calculation of the number of topologies on the set Y each of which induces the discrete topology on the set X. 
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V. I. Arnautov; A. V. Kochina. Method for constructing one-point expansions of a~topology on a~finite set and its applications. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2010), pp. 67-76. https://geodesic-test.mathdoc.fr/item/BASM_2010_3_a7/

[1] Gustavo N., Rubisno O., “Sobre el numero de topologias en un conjunto finito”, Boletin de Matematicas. Nueva Serie, 13:2 (2006), 136–158 | MR | Zbl

[2] The on-Line Encyclopedia of Integer Sequences!, number of topologies http://www.research.att.com

[3] Birkhoff G., The Theory of lattices, Nauka, Moscow, 1984 (Russian) | MR