Dichotomies pour les espaces de suites réelles

Pierre Casevitz¹

¹

Fundamenta Mathematicae, Tome 165 (2000) no. 3, pp. 249-284.

Résumé

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 ${ℝ}^{\omega }$ of real sequences, i.e., subspaces such that $[y=(y_n{)}_n\in X$ and ∀n, $|x_n|\le |y_n|]\Rightarrow x=(x_n{)}_n\in X$. If such an X is analytic as a subset of ${ℝ}^{\omega }$, 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 ${\ell }_{\infty }$ 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

Détail

DOI : 10.4064/fm-165-3-249-284

Mots-clés : Borel complexity, subspaces of real sequences, topology of subspaces of real sequences, Polishable spaces, dichotomy theorems, Borel equivalence relations