# Espaces de suites réelles complètement métrisables

Fundamenta Mathematicae (2001)

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

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## Abstract

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Let X be an hereditary subspace of the Polish space ${ℝ}^{\omega }$ 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 ${ℝ}^{\omega }$. In particular, if X is separable, then X = E. ∙ If X is an hereditary space, analytic as a subset of ${ℝ}^{\omega }$, 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 ${ℝ}^{\omega }$, 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.

## How to cite

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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
ER -

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