Countable Choice and Compactness

TL;DR

This paper investigates the role of countable choice in set theory without the full axiom of choice, establishing compactness results for Banach space balls and subsets of l^p spaces under weaker choice axioms.

Contribution

It demonstrates that certain compactness properties in Banach spaces and l^p spaces depend on countable choice or weaker choice principles within ZF set theory.

Findings

Closed unit ball of uniformly convex Banach space is convex-compact in the convex topology under ZF+AC(N).

Compactness of specific subsets of l^p spaces depends on AC(N) or DC.

Compactness results hold without full Axiom of Choice, relying on weaker choice principles.

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 l^p(I), we show that AC(N) (resp. DC, the axiom of Dependent Choices) implies the compactness of F.