Geodesic
A~class of groups in which all unconditionally closed sets are algebraic
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 8, pp. 217-222.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that, in any subgroup of a direct product of countable groups, the property of being an unconditionally closed set coincides with that of being an algebraic set. 
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O. V. Sipacheva. A~class of groups in which all unconditionally closed sets are algebraic. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 8, pp. 217-222. http://geodesic.mathdoc.fr/item/FPM_2006_12_8_a11/

[1] Markov A. A., “O bezuslovno zamknutykh mnozhestvakh”, DAN SSSR, 44:5 (1944), 196–197

[2] Markov A. A., “O svobodnykh topologicheskikh gruppakh”, Izv. AN SSSR. Ser. mat., 9:1 (1945), 3–64 | MR | Zbl

[3] Markov A. A., “O bezuslovno zamknutykh mnozhestvakh”, Mat. sb., 18:1 (1946), 3–28 | MR | Zbl

[4] Kaplansky I., Infinite Abelian Groups, Univ. of Michigan Press, Ann Arbor, 1954 | MR | Zbl

[5] Sipacheva O. V., Consistent solution of Markov's problem about algebraic sets, arXiv:math.GR/0605558