Espaces de suites réelles complètement métrisables

Pierre Casevitz¹

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

Fundamenta Mathematicae, Tome 168 (2001) no. 3, pp. 199-235

Résumé

Let $X$ be an hereditary subspace of the Polish space ${\mathbb R}^\omega $ of real sequences, i.e. a subspace such that $[x=(x_n)_n\in X$ and $\forall n$, $|y_n|\leq |x_n|]\Rightarrow y=(y_n)_n\in X$. Does $X$ admit a complete metric compatible with its vector structure? We have two results: $\bullet $ If such an $X$ has a complete metric $\delta $, there exists a unique pair $(E,F)$ of hereditary subspaces with $E\subseteq X\subseteq F$, $(E,\delta )$ 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 (${\bf \Pi }_3^0$ or ${\bf \Sigma }_2^0$) in ${\mathbb R}^\omega $. In particular, if $X$ is separable, then $X=E$. $\bullet $ If $X$ is an hereditary space, analytic as a subset of ${\mathbb R}^\omega $, we can find a subspace of $X$ strongly isomorphic to the space $c_{00}$ of finite sequences, or we can find a pair $(E,F)$ and a metric with the same properties around $X$. If $X$ is ${\bf \Sigma }_3^0$ in ${\mathbb 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.

Détail

DOI : 10.4064/fm168-3-1

Mots-clés : hereditary subspace polish space mathbb omega real sequences subspace forall leq rightarrow does admit complete metric compatible its vector structure have results bullet has complete metric delta there exists unique pair hereditary subspaces subseteq subseteq delta complete separable complete maximal strong sense metrics have simple form spaces borel sigma mathbb omega particular separable bullet hereditary space analytic subset mathbb omega subspace strongly isomorphic space finite sequences pair metric properties around sigma mathbb omega get complete trichotomy describing possible topologies which makes precise result general there examples various situations