Abstract

We introduce the relation of almost-reduction in an arbitrary topological Ramsey space ℛ as a generalization of the relation of almost-inclusion on ℕ[∞]. This leads us to a type of ultrafilter [MATHEMATICAL SCRIPT CAPITAL U] ⊆ 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 [MATHEMATICAL SCRIPT CAPITAL U] and the Ramsey space ℛ. For example, we prove that the open coloring axiom holds on L [[MATHEMATICAL SCRIPT CAPITAL U]], extending therefore the result from [3] which gives the same conclusion for the Ramsey space ℕ[∞]