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Espaces de suites réelles complètement métrisables

Pierre Casevitz

Fundamenta Mathematicae
(2001)

- Volume: 168, Issue: 3, page 199-235
- ISSN: 0016-2736

Let X be an hereditary subspace of the Polish space ${\mathbb{R}}^{\omega}$ 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 ${\mathbb{R}}^{\omega}$. In particular, if X is separable, then X = E.
∙ If X is an hereditary space, analytic as a subset of ${\mathbb{R}}^{\omega}$, 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 ${\mathbb{R}}^{\omega}$, 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.
Pierre Casevitz. "Espaces de suites réelles complètement métrisables." Fundamenta Mathematicae 168.3 (2001): 199-235. <http://eudml.org/doc/282043>.

@article{PierreCasevitz2001,

author = {Pierre Casevitz},

journal = {Fundamenta Mathematicae},

language = {fre},

number = {3},

pages = {199-235},

title = {Espaces de suites réelles complètement métrisables},

url = {http://eudml.org/doc/282043},

volume = {168},

year = {2001},

}

TY - JOUR

AU - Pierre Casevitz

TI - Espaces de suites réelles complètement métrisables

JO - Fundamenta Mathematicae

PY - 2001

VL - 168

IS - 3

SP - 199

EP - 235

LA - fre

UR - http://eudml.org/doc/282043

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