https://hal.univ-reunion.fr/hal-03426458Morillon, MarianneMarianneMorillonLIM - Laboratoire d'Informatique et de Mathématiques - UR - Université de La RéunionHyperplanes in matroids ans the Axiom of ChoiceHAL CCSD2021[MATH.MATH-LO] Mathematics [math]/Logic [math.LO]MORILLON, Marianne2021-11-12 11:56:132022-01-21 03:28:132021-11-23 13:32:24enPreprints, Working Papers, ...application/pdf1We show that in set-theory without the axiom of choice ZF, the statement sH: Every proper closed subset of a nitary matroid is the intersection of hyperplanes including it implies AC fin , the axiom of choice for (nonempty) nite sets. We also provide an equivalent of the statement AC fin in terms of graphic matroids. Several open questions stay open in ZF, for example: does sH imply the Axiom of Choice?