# Tukey order on sets of compact subsets of topological spaces

Mamatelashvili, Ana (2014) Tukey order on sets of compact subsets of topological spaces. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## Abstract

A 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.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
Mamatelashvili, Anaanm137@pitt.eduANM137
ETD Committee:
Committee ChairGartside, P.M.gartside@pitt.eduGARTSIDE
Committee MemberHeath, R.W.rwheath@pitt.eduRWHEATH
Committee MemberKnight, R.W.Robin.Knight@maths.ox.ac.uk
Committee MemberLennard, C.lennard@pitt.eduLENNARD
Date: 24 September 2014
Date Type: Publication
Defense Date: 9 July 2014
Approval Date: 24 September 2014
Submission Date: 18 June 2014
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 140
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: antichain, calibre, cofinal, compact, continuum, embedding, function space, graph, Lindel\"of $\Sigma$, metrizable, $n$-arc connected, $n$-strongly arc connected, partial order, relative Tukey order, separable, stationary, Tukey order, unbounded.
Date Deposited: 24 Sep 2014 15:02