Tukey order on sets of compact subsets of topological spaces

摘要

A partially ordered set (poset), $Q$, is a emph{Tukey quotient} of a poset, $P$, written $Pgeq_T Q$, if there exists a map, a emph{Tukey quotient}, $phi : Po Q$ such that for any cofinal subset $C$ of $P$ the image, $phi(C)$, is cofinal in $Q$. Two posets are emph{Tukey equivalent} if they are Tukey quotients of each other. Given a collection of posets, $mathcal{P}$, the relation $leq_T$ is a partial order. The Tukey structure of $mathcal{P}$ has been intensively studied for various instances of $mathcal{P}$ [13, 14, 48, 53, 58]. Here we investigate the Tukey structure of collections of posets naturally arising in Topology. ududFor a space $X$, let $mathcal{K}(X)$ be the poset of all compact subsets of $X$, ordered by inclusion, and let $mathit{Sub}(X)$ be the set of all homeomorphism classes of subsets of $X$. Let $mathcal{K}(mathit{Sub}(X))$ be the set of all Tukey classes of the form $[mathcal{K}(Y)]_T$, where $Y in mathit{Sub}(X)$. The main purpose of this work is to study order properties of $(mathcal{K}(mathit{Sub}(mathbb{R})),leq_T)$ and $(mathcal{K}(mathit{Sub}(omega_1)),leq_T)$. ududWe attack this problem using two approaches. The first approach is to study internal order properties of elements of $mathcal{K}(mathit{Sub}(mathbb{R}))$ and $mathcal{K}(mathit{Sub}(omega_1))$ that respect the Tukey order --- calibres and spectra. The second approach is more direct and studies the Tukey relation between the elements of $(mathcal{K}(mathit{Sub}(mathbb{R})),leq_T)$ and $(mathcal{K}(mathit{Sub}(omega_1)),leq_T)$. ududAs a result we show that $(mathcal{K}(mathit{Sub}(mathbb{R})),leq_T)$ has size $2^mathfrak{c}$, has no largest element, contains an antichain of maximal size, $2^mathfrak{c}$, its additivity is $mathfrak{c}^+$, its cofinality is $2^mathfrak{c}$, $mathcal{K}(mathit{Sub}(mathbb{R}))$ has calibre $(kappa, lambda, mu)$ if and only if $mu leq mathfrak{c}$ and $mathfrak{c}^+$ is the largest cardinal that embeds in $mathcal{K}(mathit{Sub}(mathbb{R}))$. While the size and the existence of large antichains of $mathcal{K}(mathit{Sub}(omega_1))$ have already been established in [58], we determine special classes of $mathcal{K}(mathit{Sub}(omega_1))$ and the relation between these classes and the elements of $mathcal{K}(mathit{Sub}(mathbb{R}))$. ududFinally, we explore connections of the Tukey order with function spaces and the Lindel"of $Sigma$ property, which require giving the Tukey order more flexibility and larger scope. Hence we develop the emph{relative} Tukey order and present applications of relative versions of results on $(mathcal{K}(mathit{Sub}(mathbb{R})),leq_T)$ and $(mathcal{K}(mathit{Sub}(omega_1)),leq_T)$ to function spaces.