Continuous Cofinal Maps on Ultrafilters

An ultrafilter U on a countable base set B has continuous Tukey reductions if whenever an ultrafilter V is Tukey reducible to U , then every monotone cofinal map f : U → V is continuous with respect to the Cantor topology, when restricted to some cofinal subset of U . We show that the slightly stronger property of having basic Tukey reductions is inherited under Tukey reducibility. It follows that the class of ultrafilters Tukey reducible to any p-point has continuous Tukey reductions. We also prove that every countable iterate of Fubini products of p-points has finitary Tukey reductions. The proof makes use of the association between countable iterates of Fubini products of p-points and ultrafilters generated by ~ U-trees on some front B. The finitary Tukey reductions are in fact continuous when viewed on the space of P(B̂) with the Cantor topology, where B̂ is the tree of all initial segments of members of the front B.