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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 ℓ 1(I) (resp. such that F ⊆ c 0(I)), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC ℕ) implies that F is compact. This enhances previous results where AC ℕ (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 ℓ 2 (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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