Countable Choice and Compactness
Functional Analysis
2008-12-18 v1 General Topology
Logic
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.
Cite
@article{arxiv.0803.3131,
title = {Countable Choice and Compactness},
author = {Marianne Morillon},
journal= {arXiv preprint arXiv:0803.3131},
year = {2008}
}