Countable choice and compactness

We work in set-theory without choice ZF. Denoting by AC ( N ) the countable axiom of choice, we show in ZF + AC ( N ) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p ⩾ 1 (respectively p = 0 ), and some closed subset F of [ 0 , 1 ] I which is a bounded subset of ℓ p ( I ) , we show that AC ( N ) (respectively DC, the axiom of Dependent Choices) implies the compactness of F.