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 t...

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are n...

One partially ordered set, Q, is a Tukey quotient of another, P – denoted P?TQ if there map ?:P?Q carrying cofinal sets to Q. Let X be space and denote by K(X) the set compact subsets X, inclusion. For certain separable metrizable spaces M, upper lower bounds K(M) are calculated. Results on invariants K(M)'s deduced. The structure all under ?T investigated. Particular emphasis placed position w...

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell’s Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known ca...

It has been twenty five years since John Tukey published the preliminary limited edition of Exploratory Data Analysis. All the standard texts have been out for ten or more years: Tukey (1977), Mosteller and Tukey (1977), Mosteller, Hoaglin, and Tukey (1983, 1985). So how is it that these important statistical ideas are only now being used to improve the way we collect business statistics, and w...