A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections

P. Holický; Miroslav Zelený

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 3, page 191-202
  • ISSN: 0016-2736

Abstract

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Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then f - 1 ( y ) is a K σ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of a Luzin space under a closed continuous mapping is a Luzin space. The method is based on a parametrized version of a Hurewicz type theorem and on the use of the Jankov-von Neumann selection theorem.

How to cite

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Holický, P., and Zelený, Miroslav. "A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections." Fundamenta Mathematicae 165.3 (2000): 191-202. <http://eudml.org/doc/212466>.

@article{Holický2000,
abstract = {Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then $f^\{-1\}(y)$ is a $K_σ$ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of a Luzin space under a closed continuous mapping is a Luzin space. The method is based on a parametrized version of a Hurewicz type theorem and on the use of the Jankov-von Neumann selection theorem.},
author = {Holický, P., Zelený, Miroslav},
journal = {Fundamenta Mathematicae},
keywords = {$K_σ$ sections; Borel bimeasurability},
language = {eng},
number = {3},
pages = {191-202},
title = {A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections},
url = {http://eudml.org/doc/212466},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Holický, P.
AU - Zelený, Miroslav
TI - A converse of the Arsenin–Kunugui theorem on Borel sets with σ-compact sections
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 3
SP - 191
EP - 202
AB - Let f be a Borel measurable mapping of a Luzin (i.e. absolute Borel metric) space L onto a metric space M such that f(F) is a Borel subset of M if F is closed in L. We show that then $f^{-1}(y)$ is a $K_σ$ set for all except countably many y ∈ M, that M is also Luzin, and that the Borel classes of the sets f(F), F closed in L, are bounded by a fixed countable ordinal. This gives a converse of the classical theorem of Arsenin and Kunugui. As a particular case we get Taĭmanov’s theorem saying that the image of a Luzin space under a closed continuous mapping is a Luzin space. The method is based on a parametrized version of a Hurewicz type theorem and on the use of the Jankov-von Neumann selection theorem.
LA - eng
KW - $K_σ$ sections; Borel bimeasurability
UR - http://eudml.org/doc/212466
ER -

References

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  1. [1] C. Dellacherie, Un cours sur les ensembles analytiques, in: Analytic Sets, Academic Press, London, 1980, 183-316. 
  2. [2] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995. 
  3. [3] K. Kuratowski, Topology, Vol. I, Academic Press, New York, 1966. 
  4. [4] A. Louveau and J. Saint-Raymond, Borel classes and closed games, Trans. Amer. Math. Soc. 304 (1987), 431-467. Zbl0655.04001
  5. [5] R. D. Mauldin, Bimeasurable functions, Proc. Amer. Math. Soc. 83 (1981), 369-370. Zbl0483.54026
  6. [6] R. Pol, Some remarks about measurable parametrizations, ibid. 93 (1985), 628-632. Zbl0609.28006
  7. [7] R. Purves, Bimeasurable functions, Fund. Math. 58 (1966), 149-158. Zbl0143.07101
  8. [8] J. Saint-Raymond, Boréliens à coupes K σ , Bull. Soc. Math. France 104 (1976), 389-400. 
  9. [9] S. M. Srivastava, A Course on Borel Sets, Springer, New York, 1998. Zbl0903.28001
  10. [10] A. D. Taĭmanov, On closed mappings I, Mat. Sb. 36 (1955), 349-352 (in Russian). 

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