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
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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/

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