Parametrizing the abstract Ellentuck theorem

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.