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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Paramétrisations boréliennes


Author: Gabriel Debs
Journal: Proc. Amer. Math. Soc. 94 (1985), 675-681
MSC: Primary 28A20; Secondary 04A15, 54H05
MathSciNet review: 792282
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Abstract: Given a Borel subset $ B$ of the product $ X \times Y$ of two Polish spaces $ X$ and $ Y$ such that $ \{ x \in X:B(x) \cap V \ne \emptyset \} $ is Borel for any open subset $ V$ of $ Y$, then:

(1) If for any $ x \in X$ the section $ B(x)$ is a dense in itself $ {G_\delta }$-subset of $ Y$, we prove that there exists a Borel isomorphism $ f:X \times {{\mathbf{N}}^{\mathbf{N}}} \to B$ such that $ f(x, \cdot )$ is one-to-one and continuous from $ {{\mathbf{N}}^{\mathbf{N}}}$ onto $ B(x)$.

(2) If for any $ x \in X$, $ \overline {B(x)} $ is a 0-dimensional compact space and $ B(x)$ is a dense in itself $ {G_\delta }$, we prove that $ f$ may be chosen such that $ f(x, \cdot )$ is a homeomorphism from $ {{\mathbf{N}}^{\mathbf{N}}}$ onto $ G(x)$ for any $ x \in X$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0792282-3
PII: S 0002-9939(1985)0792282-3
Article copyright: © Copyright 1985 American Mathematical Society