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Dichotomies pour les espaces de suites réelles

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Casevitz P.

Fundamenta Mathematicae

2000

tom 165

nr 3

249-284

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

Espaces de suites réelles complètement métrisables

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Casevitz P.

Fundamenta Mathematicae

2001

tom 168

nr 3

199-235

Let X be an hereditary subspace of the Polish space $ℝ^{ω}$ of real sequences, i.e. a subspace such that [x = (xₙ)ₙ ∈ X and ∀n, |yₙ| ≤ |xₙ|] ⇒ y = (yₙ)ₙ ∈ X. Does X admit a complete metric compatible with its vector structure? We have two results: ∙ If such an X has a complete metric δ, there exists a unique pair (E,F) of hereditary subspaces with E ⊆ X ⊆ F, (E,δ) complete separable, and F complete maximal in a strong sense. On E and F, the metrics have a simple form, and the spaces E are Borel (Π₃⁰ or Σ₂⁰) in $ℝ^{ω}$. In particular, if X is separable, then X = E. ∙ If X is an hereditary space, analytic as a subset of $ℝ^{ω}$, we can find a subspace of X strongly isomorphic to the space c₀₀ of finite sequences, or we can find a pair (E,F) and a metric with the same properties around X. If X is Σ₃⁰ in $ℝ^{ω}$, we get a complete trichotomy describing the possible topologies of X, which makes precise a result of [C], but for general X's, there are examples of various situations.

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2 Fundamenta Mathematicae

2 Casevitz P.

1 2001

1 2000

Wyniki wyszukiwania

Dichotomies pour les espaces de suites réelles

Espaces de suites réelles complètement métrisables