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

Fundamenta Mathematicae (2000)

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

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topHolický, 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

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

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