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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Paramétrisations boréliennes
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by Gabriel Debs PDF
Proc. Amer. Math. Soc. 94 (1985), 675-681 Request permission

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 $x \in X$.
References
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 94 (1985), 675-681
  • MSC: Primary 28A20; Secondary 04A15, 54H05
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0792282-3
  • MathSciNet review: 792282