Dichotomies pour les espaces de suites réelles

Pierre Casevitz

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 3, page 249-284
  • ISSN: 0016-2736

Abstract

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There is a general conjecture, the dichotomy (C) about Borel equivalence relations E: (i) E is Borel reducible to the equivalence relation E G X 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

How to cite

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Casevitz, Pierre. "Dichotomies pour les espaces de suites réelles." Fundamenta Mathematicae 165.3 (2000): 249-284. <http://eudml.org/doc/212469>.

@article{Casevitz2000,
author = {Casevitz, Pierre},
journal = {Fundamenta Mathematicae},
keywords = {Borel complexity; subspaces of real sequences; topology of subspaces of real sequences; Polishable spaces; dichotomy theorems; Borel equivalence relations; space of real sequences; Polishable space; Polish space; Polish group},
language = {fre},
number = {3},
pages = {249-284},
title = {Dichotomies pour les espaces de suites réelles},
url = {http://eudml.org/doc/212469},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Casevitz, Pierre
TI - Dichotomies pour les espaces de suites réelles
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 3
SP - 249
EP - 284
LA - fre
KW - Borel complexity; subspaces of real sequences; topology of subspaces of real sequences; Polishable spaces; dichotomy theorems; Borel equivalence relations; space of real sequences; Polishable space; Polish space; Polish group
UR - http://eudml.org/doc/212469
ER -

References

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  1. [C] P. Casevitz, Espaces héréditaires complètement métrisables, Fund. Math., à paraître. 
  2. [K] A. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995. Zbl0819.04002
  3. [K-L] A. S. Kechris and A. Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), 215-242. Zbl0865.03039
  4. [K-L-W] A. S. Kechris, A. Louveau and W. H. Woodin, The structure of σ-ideals of compact sets, Trans. Amer. Math. Soc. 301 (1987), 263-288. Zbl0633.03043
  5. [M] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980. 
  6. [Sc] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York, 1974. Zbl0296.47023
  7. [So] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51-72. Zbl0932.03060
  8. [T] M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), 13-43. Zbl0435.46023

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