  After calculating the Dushnik-Miller dimension of Minkowski spaces to be
countable infinity, we define a novel notion of dimension for ordered spaces
recovering the correct manifold dimension and obtain a corresponding
obstruction for the existence of injective monotonous maps between Lorentzian
length spaces. Furthermore we induce metrics on Cauchy subsets, relate
respective Hausdorff dimensions, prove existence of rushing Cauchy functions
with a given Cauchy zero locus and consider collapse phenomena in this setting.
