# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Dichotomies pour les espaces de suites réelles

### Tom 165 / 2000

Fundamenta Mathematicae 165 (2000), 249-284 DOI: 10.4064/fm-165-3-249-284

#### Streszczenie

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

#### Autorzy

• Pierre Casevitz

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