Tukey order on sets of compact subsets of topological spacesMamatelashvili, Ana (2014) Tukey order on sets of compact subsets of topological spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractA partially ordered set (poset), $Q$, is a \emph{Tukey quotient} of a poset, $P$, written $P\geq_T Q$, if there exists a map, a \emph{Tukey quotient}, $\phi : P\to Q$ 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. For 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 $(\mathcal{K}(\mathit{Sub}(\omega_1)),\leq_T)$. We 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 $(\mathcal{K}(\mathit{Sub}(\omega_1)),\leq_T)$. As 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 $\mathcal{K}(\mathit{Sub}(\mathbb{R}))$. Finally, 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. Share
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