Espaces de suites réelles complètement métrisables

Pierre Casevitz

Fundamenta Mathematicae (2001)

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

Abstract

top
Let X be an hereditary subspace of the Polish space 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 ω . In particular, if X is separable, then X = E. ∙ If X is an hereditary space, analytic as a subset of 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 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.

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.