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

• Autorzy: P. Holický , Miroslav Zelený
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

$K_σ$ sections; Borel bimeasurability

Kategorie tematyczne

• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
• 26A21: Classification of real functions; Baire classification of sets and functions
• 28A05: Classes of sets (Borel fields, σ -rings, etc.), measurable sets, Suslin sets, analytic sets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 165
• Numer: 3
• Strony: 191-202

Daty

wydano

2000

otrzymano

1999-05-27

poprawiono

2000-03-16

Twórcy

autor

P. Holický

• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic,

autor

Miroslav Zelený

• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic,

Bibliografia

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• [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.
• [5] R. D. Mauldin, Bimeasurable functions, Proc. Amer. Math. Soc. 83 (1981), 369-370.
• [6] R. Pol, Some remarks about measurable parametrizations, ibid. 93 (1985), 628-632.
• [7] R. Purves, Bimeasurable functions, Fund. Math. 58 (1966), 149-158.
• [8] J. Saint-Raymond, Boréliens à coupes $K_σ$, Bull. Soc. Math. France 104 (1976), 389-400.
• [9] S. M. Srivastava, A Course on Borel Sets, Springer, New York, 1998.
• [10] A. D. Taĭmanov, On closed mappings I, Mat. Sb. 36 (1955), 349-352 (in Russian).