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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
  • SDAD, Université de Caen, Campus II, Boulevard Maréchal Juin, 1, Esplanade de la Libération, BP 5186, F-14032 Caen Cedex, France
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