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?
Keywords: axiom of choice; finitary matroid; circuit; hyperplane; graph
DOI: DOI 10.14712/1213-7243.2023.010
AMS Subject Classification: 03E25 05B99