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

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

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn1447
• https://doi.org/10.4213/rm1447
• http://mi.mathnet.ru/eng/umn/v60/i4/p123

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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
2. A. V. Ostrovsky, “Borel sets as sums of canonical elements”, Dokl. Math., 75:2 (2007), 213–217
3. Spurný J., “Borel sets and functions in topological spaces”, Acta Math. Hungar., 129:1-2 (2010), 47–69
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