Uniform Eberlein spaces and the finite axiom of choice

Marianne Morillon (1)

1 ERMIT - Equipe Réunionnaise de Mathématiques et Informatique Théorique

We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em resp.} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ({\em resp.} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of $\ell^2(I)$ is weakly compact (in $\ZF$).

Mots clés: Axiom of Choice; product topology; compactness; Eberlein spaces; uniform Eberlein spaces

Domaines: Analyse fonctionnelle [math.FA]; Logique [math.LO]; Topologie générale [math.GN]

Marianne Morillon. Uniform Eberlein spaces and the finite axiom of choice. 2008. ⟨hal-00268797⟩