Hyperplanes in matroids and the axiom of choice

Abstract

We show that in set theory without the axiom of choice ZF, the statement sH: ``Every proper closed subset of a finitary matroid is the intersection of hyperplanes including it'' implies AC$^{\rm fin}$, the axiom of choice for (nonempty) finite sets. We also provide an equivalent of the statement AC$^{\rm fin}$ in terms of ``graphic'' matroids. Several open questions stay open in ZF, for example: does sH imply the axiom of choice?