The Tukey Order and Subsets of ω1

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ϕ : 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 K(X)$\mathcal {K}(X)$ the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of K(S)$\mathcal {K}(S)$ corresponding to various subspaces S of ω1, their Tukey invariants, and hence the Tukey relations between them. It is shown that ωω is a strict Tukey quotient of Σ(ωω1)${\Sigma }(\omega ^{\omega _{1}})$ 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 K(S)$\mathcal {K}(S)$, where S is a subspace of ω1, and K(M)$\mathcal {K}(M)$, where M is a separable metrizable space, are revealed. Applications are given to function spaces.