The Tukey Order and Subsets of ω 1

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map I center dot : P -> Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of corresponding to various subspaces S of omega (1), their Tukey invariants, and hence the Tukey relations between them. It is shown that omega (omega) is a strict Tukey quotient of and thus we distinguish between two Tukey classes out of Isbell's ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394-416, 1972). The relationships between Tukey equivalence classes of , where S is a subspace of omega (1), and , where M is a separable metrizable space, are revealed. Applications are given to function spaces.