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 I[0,1] which is a bounded subset of ℓp(I), we show that AC(N) (respectively DC, the axiom of Dependent Choices) implies the compactness of F.