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Paramétrisations boréliennes


Author: Gabriel Debs
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
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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$.


References [Enhancements On Off] (What's this?)

  • [1] D. Cenzer et R. D. Mauldin, Measurable parametrizations and selections, Trans. Amer. Math. Soc. 245 (1978), 399-408. MR 511418 (80i:28010)
  • [2] J. P. R. Christensen, Topologie and borel structure, North-Holland Math. Studies, No. 10, North-Holland, Amsterdam, 1974.
  • [3] G. Debs, Sélections d'une multi-applications à valeurs $ {\mathcal{G}_\delta }$, Bull. Acad. Roy. Belg. 65 (1979), 211-216. MR 559141 (81e:54018)
  • [4] R. D. Mauldin et H. Sarbadhikari, Continusus one-to-one parametrizations, Bull. Sci. Math. 105 (1981), 435-444. MR 640152 (83i:28015)
  • [5] J. Saint Raymond, Boréliens à coupes $ {K_\sigma }$, Bull. Soc. Math. France 104 (1976), 389-400. MR 0433418 (55:6394)
  • [6] V. V. Srivasta, Measurable parametrizations of sets in product spaces, Trans. Amer. Math. Soc. 270 (1982), 537-556. MR 645329 (83e:54035)
  • [7] C. A. Rogers (ed.), Analytic sets, Academic Press, London, 1980. MR 608794 (82m:03063)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0792282-3
Article copyright: © Copyright 1985 American Mathematical Society

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