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Dichotomies pour les espaces de suites réelles

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There is a general conjecture, the dichotomy (C) about Borel equivalence relations E: (i) E is Borel reducible to the equivalence relation $E^X_G$ where X is a Polish space, and a Polish group acting continuously on X; or (ii) a canonical relation $E_1$ is Borel reducible to E. (C) is only proved for special cases as in [So].
 In this paper we make a contribution to the study of (C): a stronger conjecture is true for hereditary subspaces of the Polish space $ℝ^ω$ of real sequences, i.e., subspaces such that $[y=(y_n)_n ∈ X$ and ∀n, $|x_n| ≤ |y_n|] ⇒ x=(x_n)_n ∈ X$. If such an X is analytic as a subset of $ℝ^ω$, then either X is Polishable as a vector subspace, or X admits a subspace strongly isomorphic to the space $c_{00}$ of finite sequences, or to the space $ℓ_∞$ of bounded sequences.
 When X is Polishable, the metrics have a very simple form as in the case studied in [So], which allows us to study precisely the properties of those X's
Twórcy
  • SDAD, Université de Caen, Campus II, Boulevard Maréchal Juin, 1, Esplanade de la Libération, BP 5186, F-14032 Caen Cedex, France, asevitz@math.unicaen.fr
Bibliografia
  • [C] P. Casevitz, Espaces héréditaires complètement métrisables, Fund. Math., à paraître.
  • [K] A. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.
  • [K-L] A. S. Kechris and A. Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), 215-242.
  • [K-L-W] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288.
  • [M] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
  • [Sc] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York, 1974.
  • [So] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51-72.
  • [T] M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), 13-43.
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Bibliografia
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bwmeta1.element.bwnjournal-article-fmv165i3p249bwm
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