Countable Choice and Compactness
Abstract
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 greater or equal to 1 (resp. . p = 0), and some closed subset F of [0, 1]I which is a bounded subset of lp(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.
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