Uniform Eberlein spaces and the finite axiom of choice

TL;DR

This paper investigates the compactness of certain subsets in product spaces under set theory without choice, showing that specific forms of the axiom of choice imply compactness results that were previously known only under stronger assumptions.

Contribution

It demonstrates that weaker forms of the axiom of choice suffice to establish compactness of bounded subsets in certain infinite-dimensional spaces, improving upon earlier results requiring stronger choice axioms.

Findings

Countable choice for finite subsets implies compactness of bounded sets in $eta$-spaces.

Weak compactness of the closed unit ball of $oldsymbol{ ext{ell}}^2(I)$ in $ ext{ZF}$ when $I$ is linearly orderable.

Enhanced results under $ ext{ZF}$ without full choice axioms.

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 $\ZF$).