Dichotomies pour les espaces de suites réelles

• Autorzy: Pierre Casevitz
• Języki publikacji: FR

Abstrakty

EN

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

Słowa kluczowe

EN

Borel complexity; subspaces of real sequences; topology of subspaces of real sequences; Polishable spaces; dichotomy theorems; Borel equivalence relations

Kategorie tematyczne

• 54D65: Separability
• 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
• 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
• 54E52: Baire category, Baire spaces
• 06F30: Topological lattices, order topologies
• 46A45: Sequence spaces (including K\"othe sequence spaces)
• 03E15: Descriptive set theory
• 54D55: Sequential spaces
• 46B40: Ordered normed spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2000
• Tom: 165
• Numer: 3
• Strony: 249-284

Daty

wydano

2000

otrzymano

1999-12-17

Twórcy

autor

Pierre Casevitz

• SDAD, Université de Caen, Campus II, Boulevard Maréchal Juin, 1, Esplanade de la Libération, BP 5186, F-14032 Caen Cedex, France

Bibliografia

• [C] P. Casevitz, Espaces héréditaires complètement métrisables, Fund. Math., à paraître.
• [K] A. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.
• [K-L] A. S. Kechris and A. Louveau, The classification of hypersmooth Borel equivalence relations, J. Amer. Math. Soc. 10 (1997), 215-242.
• [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.
• [M] Y. N. Moschovakis, Descriptive Set Theory, North-Holland, Amsterdam, 1980.
• [Sc] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York, 1974.
• [So] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51-72.
• [T] M. Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), 13-43.