Qualitative reasoning about commonsense space often involves entities of different dimensions. We present a weak axiomatization of multidimensional qualitative space based on `relative dimension' and dimension-independent `containment' which suffice to define basic dimension-dependent mereotopological relations. We show the relationships to other meoreotopologies and to incidence geometry. The extension with betweenness, a primitive of relative position, results in a first-order theory that qualitatively abstracts ordered incidence geometry.