The Tukey Order and Subsets of $\omega_1$

One partially ordered set, $Q$, is a Tukey quotient of another, $P$, if there is a map $\phi : P \to Q$ 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 $\mathcal{K}(X)$ the set of compact subsets of $X$, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of $\mathcal{K}(S)$ 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 $\Sigma(\omega^{\omega_1})$ and thus we distinguish between two Tukey classes out of Isbell's ten partially ordered sets. The relationships between Tukey equivalence classes of $\mathcal{K}(S)$, where $S$ is a subspace of $\omega_1$, and $\mathcal{K}(M)$, where $M$ is a separable metrizable space, are revealed. Applications are given to function spaces.