Uniform Eberlein spaces and the finite axiom of choice

Abstract

We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1{]}^I$ which is a bounded subset of ${\ell }^1(I)$ ({\em resp.} such that $F\subseteq {\ell }^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em resp.} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ({\em resp.} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of ${\ell }^2(I)$ is weakly compact (in $\ZF$).