COUNTABLE CHOICE AND COMPACTNESS

Marianne Morillon (1)

1 ERMIT - Equipe Réunionnaise de Mathématiques et Informatique Théorique

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]^I which is a bounded subset of l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.

Mots clés: Axiom of Choice; Banach space; weak compactness; Hahn-Banach; uniformly convex; Axiom of Choice.

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

Marianne Morillon. COUNTABLE CHOICE AND COMPACTNESS. 2008. ⟨hal-00266108⟩