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Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Marianne Morillon
Marianne Morillon*
Affiliation:
Université de la Réunion, Parc Technologique Universitaire, Ermit, Département de Mathématiques et Informatique, Bâtiment 2, 2 Rue Joseph Wetzell, 97490 Sainte-Clotilde, France, E-mail: mar@univ-reunion.fr, URL: http://personnel.univ-reunion.fr/mar
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Abstract

We work in set-theory without choice ZF. A set is countable if it is finite or equipotent with ℕ. Given a closed subset F of [0, 1]I which is a bounded subset of ℓ1(I) (resp. such that F ⊆ c0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice ACℕ) implies that F is compact. This enhances previous results where ACℕ (resp. the axiom of Dependent Choices) was required. If I is linearly orderable (for example I = ℝ), then, in ZF, the closed unit ball of the Hilbert space ℓ2 (I) is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of is not provable in ZF.

Keywords

Axiom of Choiceproduct topologycompactnesssequential compactness
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 1 , March 2010 , pp. 255 - 268
Copyright
Copyright © Association for Symbolic Logic 2010

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