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# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

Online
ISSN
1572-9176
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# Images of Bernstein sets via continuous functions

Jacek Cichoń
• Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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/ Michał Morayne
• Corresponding author
• Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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• Other articles by this author:
/ Robert Rałowski
• Department of Computer Science, Faculty of Fundamental Problems of Technology, Wrocław University of Science and Technology, 50-370 Wrocław, Poland
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Published Online: 2019-08-28 | DOI: https://doi.org/10.1515/gmj-2019-2041

## Abstract

We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function $f:ℝ\to ℝ$ that maps every Bernstein subset of $ℝ$ onto the whole real line. This gives the positive answer to a question of Osipov.

Keywords: Bernstein set; Vitali set; continuous function

MSC 2010: 03E75; 03E15; 54C05; 26A99

Dedicated to Professor Alexander B. Kharazishvili on the occasion of his 70th birthday

## References

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M. Beriashvili, On some paradoxical subsets of the real line, Georgian Int. J. Sci. Technol. 6 (2014), no. 4, 265–275. Google Scholar

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M. Beriashvili, Measurability properties of certain paradoxical subsets of the real line, Georgian Math. J. 23 (2016), no. 1, 25–32.

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H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1967), 241–249.

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M. Kysiak, Bernstein sets with algebraic properties, Cent. Eur. J. Math. 7 (2009), no. 4, 725–731.

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J. Mycielski, Algebraic independence and measure, Fund. Math. 61 (1967), 165–169.

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S. Saks, Theory of the Integral, Dover, New York, 1964. Google Scholar

Accepted: 2019-01-22

Published Online: 2019-08-28

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

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© 2019 Walter de Gruyter GmbH, Berlin/Boston.