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.
"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