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