For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$, and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa.

We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces.

Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $-pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$.

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph _n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the class $\mathcal D_{\aleph _n}$ of directed sets of cardinality $\leq \aleph _n$ contains an isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

We investigate the Tukey order in the class of $F_{\sigma }$ ideals of subsets of $\omega $ . We show that no nontrivial $F_{\sigma }$ ideal is Tukey below a $G_{\delta }$ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets.

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map (a Tukey quotient) $\phi :P \to Q$ carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients of each other are said to be Tukey equivalent. Let ${\cal D}_{\rm{}} $ be the partially ordered set of Tukey equivalence classes of directed sets of size $ \le {\rm{}}$. It is shown that ${\cal D}_{\rm{}} $ contains an antichain of size $2^{\rm{}} $, and so has size $2^{\rm{}} $. The elements of the antichain are of the form ${\cal K}\left( M \right)$, the set of compact subsets of a separable metrizable space M, ordered by inclusion. The order structure of such ${\cal K}\left( M \right)$’s under Tukey quotients is investigated. Relative Tukey quotients are introduced. Applications are given to function spaces and to the complexity of weakly countably determined Banach spaces and Gul’ko compacta.

We study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.