Geodesic
The lower neighborhoods problem in a~space with closure and a~finitary theorem
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2011), pp. 65-70

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In this paper we study conditions under which all finitely generated subsets in a space (i.e., in a set with the closure property) have finite lower neighborhoods. We prove the finitary property of such a space. In addition, we state the Birchoff and Frink finatary theorem more precisely. Considered questions have applications to discrete functional systems with superpositions.
Keywords: space, closure property, closure property of finite character, completeness problem, expressibility problem, finitely generated classes, criterion system, lower neighborhood. 
@article{IVM_2011_2_a6,
     author = {N. G. Parvatov},
     title = {The lower neighborhoods problem in a~space with closure and a~finitary theorem},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {65--70},
     publisher = {mathdoc},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2011_2_a6/}
}
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N. G. Parvatov. The lower neighborhoods problem in a~space with closure and a~finitary theorem. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2011), pp. 65-70. http://geodesic.mathdoc.fr/item/IVM_2011_2_a6/