Source: J. Symbolic Logic
Volume 75, Issue 1
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 ZFℕ) implies
that F is compact. This enhances previous results where ZFℕ (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.
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