Uniform Eberlein Compactness and the Axiom of Choice.
by
Marianne Morillon
University of La Reunion (France)

We work in set-theory without the Axiom of Choice ZF. Given a set I, we consider the following weak forms of the Axiom of Choice:
AC^fin(I): "The set of finite non-empty subsets of X has a choice function."
T_w^fin(I): "Every sequence (F_n)_{n ∈ w} of finite discrete subsets of I has a compact product."
We endow [0, 1]^I with the product topology, and we denote by B⁺(I) the closed subspace of elements x=(x_i)_{i ∈ I} ∈ [0, 1]^I such that ∑_{i ∈ I}x_i ≤ 1. We show (in ZF) that T_w^fin(I) is equivalent to the compactness of B⁺(I). Moreover, AC^fin(I) is equivalent to the following statement: "The space B⁺(I) is closely compact." Here, a topological space is closely compact if it is compact and if the set of non-empty closed subsets has a choice function. Denoting by R the set of real numbers, it follows that B⁺(R) is closely compact, but the compactness of B⁺({0, 1}^R) is not provable in ZF.

Date received: June 30, 2008