Espaces de suites réelles complètement métrisables

Pierre Casevitz

Fundamenta Mathematicae, Tome 167 (2001), p. 199-235 / Harvested from The Polish Digital Mathematics Library

Résumé

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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Publié le : 2001-01-01
EUDML-ID : urn:eudml:doc:282043