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March 2010 Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice
Marianne Morillon
J. Symbolic Logic 75(1): 255-268 (March 2010). DOI: 10.2178/jsl/1264433919

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 ZF) implies that F is compact. This enhances previous results where ZF (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 l²(𝒫(ℝ)) is not provable in ZF.

Citation

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Marianne Morillon. "Notions of compactness for special subsets of ℝI and some weak forms of the axiom of choice." J. Symbolic Logic 75 (1) 255 - 268, March 2010. https://doi.org/10.2178/jsl/1264433919

Information

Published: March 2010
First available in Project Euclid: 25 January 2010

zbMATH: 1190.03042
MathSciNet: MR2605892
Digital Object Identifier: 10.2178/jsl/1264433919

Subjects:
Primary: 54D30 , Primary 03E25 , Secondary 54B10

Keywords: axiom of choice , compactness , product topology , Sequential compactness

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 1 • March 2010
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