Ultrafilters on metric spaces

For a subset Y⊆X and p∈X#, we denote △p(Y)=Y#∩{q∈X#:p∥q} and say that a subset S⊆X# is an ultracompanion of Y if S=△p(Y) for some p∈X#. We characterize large, thick, prethick, small, thin and asymptotically scattered spaces in terms of their ultracompanions.