Journal of Symbolic Logic

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

Marianne Morillon

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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.

Article information

J. Symbolic Logic, Volume 75, Issue 1 (2010), 255-268.

First available in Project Euclid: 25 January 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: Primary 03E25 Secondary 54B10 54D30: Compactness

Axiom of Choice product topology compactness sequential compactness


Morillon, Marianne. Notions of compactness for special subsets of ℝ I and some weak forms of the axiom of choice. J. Symbolic Logic 75 (2010), no. 1, 255--268. doi:10.2178/jsl/1264433919.

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