Geometric condition for Dependent Choice

Abstract

We provide a geometric condition which characterises when the Principle of Dependent Choice holds in a Fraenkel--Mostowski--Specker permutation model. This condition is a slight weakening of requiring the filter of groups to be closed under countable intersections. We show that this condition holds nontrivially in a new permutation model we call "the nowhere dense model" and we study its extensions to uncountable cardinals as well.