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