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## Fundamenta Mathematicae

2000 | 165 | 3 | 191-202
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
191-202
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-27
poprawiono
2000-03-16
Twórcy
autor
• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
autor
• Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic
Bibliografia
• [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.
• [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).
Typ dokumentu
Bibliografia
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