  We give a parametrization with perfect sets of the abstract Ellentuck
theorem. The main tool for achieving this goal is a sort of parametrization of
an abstract version of the Nash-Williams theorem. As corollaries, we obtain
some known classical results like the parametrized version of the Galvin-Prikry
theorem due to Miller and Todorcevic, and the parametrized version of
Ellentuck's theorem due to Pawlikowski. Also, we obtain parametized vesions of
nonclassical results such as Milliken's theorem.
