We will show that every subforcing of the basic Prikry forcing is either trivial or isomorphic to the Prikry forcing with the same ultrafilter. Let κ be a measurable cardinal and U a normal ultrafilter over κ. We will denote by P (U) the basic Prikry forcing with U . Let us recall the definition. Definition 0.1 P (U) is the set of all pairs 〈p,A〉 such that 1. p is a finite subset of κ, 2. A ∈ U...

The axioms for a topology in terms of open sets follow necessarily from the intuitive relation of this concept with ultrafilter convergence. By contrast, the intuitive relations between neighbourhood systems or closure operations on the one hand and ultrafilter convergence on the other lead only to pretopologies. Kleisli operations, previously used in categorical algebra, greatly facilitate cat...

The generic ultrafilter G2 forced by P(ω × ω)/(Fin ⊗ Fin) was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove that G2 is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each k ≥ 2, the collection of all nonprincipal ultrafilters Tukey re...

Motivated by a Tukey classification problem we develop here a new topological Ramsey space R1 that in its complexity comes immediately after the classical is a natural Ellentuck space [5]. Associated with R1 is an ultrafilter U1 which is weakly Ramsey but not Ramsey. We prove a canonization theorem for equivalence relations on fronts on R1. This is analogous to the Pudlak-Rödl Theorem canonizin...

We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Every topological Ramsey space R has an associated notion of Ramsey ultrafilter for R and selective ultrafilter for R (see [1]). If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Rams...

This paper concentrates on the existence of canonical cofinal maps of three types: continuous, generated by finitary monotone end-extension preserving maps, and generated by monotone finitary maps. The main theorems prove that every monotone cofinal map on an ultrafilter of a certain class of ultrafilters is actually canonical when restricted to some filter base. These theorems are then applied...