A notion of selective ultrafilter corresponding to topological Ramsey spaces

We introduce the relation of almost-reduction in an arbitrary topological Ramsey space R as a generalization of the relation of almostinclusion on N[∞]. This leads us to a type of ultrafilter U on the set of first approximations of the elements of R which corresponds to the well-known notion of selective ultrafilter on N. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space N[∞] to the ultrafilter U and the Ramsey space R. For example, we prove that the Open Coloring Axiom holds on L(R)[U ], extending therefore the result from [3] which gives the same conclusion for the Ramsey space N[∞].