科研之友
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摘要
Motivated by Tukey classification problems and building on work in Part 1, we develop a new hierarchy of topological Ramsey spaces R-alpha, alpha < omega 1. These spaces form a natural hierarchy of complexity, R-0 being the Ellentuck space, and for each alpha < omega 1, R alpha+1 coming immediately after R-alpha in complexity. Associated with each R-alpha is an ultrafilter U-alpha, which is Ramsey for Ra, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems for equivalence relations on fronts on Ra, 2 <= alpha <= omega 1. These form a hierarchy of extensions of the Pudlak-Rodl Theorem canonizing equivalence relations on barriers on the Ellentuck space. We then apply our Ramsey-classification theorems to completely classify all Rudin-Keisler equivalence classes of ultrafilters which are Tukey reducible to U-alpha, for each 2 <= alpha <= omega 1: Every nonprincipal ultrafilter which is Tukey reducible to U-alpha is isomorphic to a countable iteration of Fubini products of ultrafilters from among a fixed countable collection of rapid p-points. Moreover, we show that the Tukey types of nonprincipal ultrafilters Tukey reducible to U-alpha form a descending chain of rapid p-points of order type alpha + 1.
- 出版日期2015-7
