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

Marianne Morillon
Source: J. Symbolic Logic Volume 75, Issue 1 (2010), 255-268.

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.

First Page:
Primary Subjects: Primary 03E25, Secondary 54B10, 54D30
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1264433919
Digital Object Identifier: doi:10.2178/jsl/1264433919
Zentralblatt MATH identifier: 05681302
Mathematical Reviews number (MathSciNet): MR2605892

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