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Uspekhi Mat. Nauk, 2005, Volume 60, Issue 4(364), Pages 123–144 (Mi umn1447)  

This article is cited in 3 scientific papers (total in 3 papers)

On some problems of descriptive set theory in topological spaces

M. M. Choban

Tiraspol State University

Abstract: Problems concerning the structure of Borel sets, their classification, and invariance of certain properties of sets under maps of given types arose in the first half of the previous century in the works of A. Lebesgue, R. Baire, N. N. Luzin, P. S. Alexandroff, P. S. Urysohn, P. S. Novikov, L. V. Keldysh, and A. A. Lyapunov and gave rise to many investigations. In this paper some results related to questions of F. Hausdorff, Luzin, Alexandroff, Urysohn, M. Katětov, and A. H. Stone are obtained. In 1934 Hausdorff posed the problem of invariance of the property of being an absolute $B$-set (that is, a Borel set in some complete separable metric space) under open continuous maps. By a theorem of Keldysh, the answer to this question is negative in general. The present paper gives additional conditions under which the answer to Hausdorff's question is positive. Some general problems of the theory of operations on sets are also treated.

DOI: https://doi.org/10.4213/rm1447

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English version:
Russian Mathematical Surveys, 2005, 60:4, 699–719

Bibliographic databases:

UDC: 515.128+515.12
MSC: Primary 54H05; Secondary 28A05, 54C10, 54D15, 54E52
Received: 11.05.2005

Citation: M. M. Choban, “On some problems of descriptive set theory in topological spaces”, Uspekhi Mat. Nauk, 60:4(364) (2005), 123–144; Russian Math. Surveys, 60:4 (2005), 699–719

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Ostrovsky, “Maps of Borel Sets”, Proc. Steklov Inst. Math., 252 (2006), 225–247  mathnet  crossref  mathscinet
    2. A. V. Ostrovsky, “Borel sets as sums of canonical elements”, Dokl. Math., 75:2 (2007), 213–217  mathnet  crossref  mathscinet  zmath  isi  elib
    3. Spurný J., “Borel sets and functions in topological spaces”, Acta Math. Hungar., 129:1-2 (2010), 47–69  crossref  mathscinet  zmath  isi
  • Успехи математических наук Russian Mathematical Surveys
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