Branch space representations of lines
type: Item
instance of: scholarly article
title: Branch space representations of lines
zbMATH Open document ID: 1064.05042
author: Will Funk
author: David Lutzer
published in: Topology and its Applications
publication date: 2005-06-10T00:00:00Z
review text: A tree is a partially ordered set \((T,\leq_T)\) such for each \(t\in T\) the set \(T_t=\{s\in T:s\neq t,s\leq_Tt\}\) is well ordered by \(\leq _T\). By a path in \(T\) we mean a subset \(\rho\) of \(T\) that is linearly ordered by \(\leq _T\) and has the property that if \(s\leq_Tt\) and \(t\in\rho\), then \(s\in\rho\). Each non-maximal path \(\rho\) determines a node of the tree by Node\((\rho)=\{t\in T:T_t=\rho\}\). By a branch of \(T\) we mean a maximal path in \(T\) and we denote the set of all branches of \(T\) by \(B_T\). For any \(b\in B_T\) let ht\((b)\) be the order type of the well-ordered set \(b\) (ordered as a subset of \((T,\leq_T) \)). For \(\alpha< \text{ht}(b)\) let \(b(\alpha)\) be the unique point of \(b\cap T_\alpha\). For distinct \(b,c\in B_T\), there is a first ordinal \(\delta=\Delta _T(b,c)\) such that \(b(\delta )\neq c(\delta)\). Then \(b(\delta),c(\delta)\) belong to the same node \(N\) of \(T\) and we define \(b<_Bc\) if and only if in the ordering \(<_N\) chosen for \(N\), we have \(b(\delta)<_Nc(\delta)\). The branch space of the tree is the linearly ordered set \((B_T,\leq_B)\). In this paper the authors investigate representing linearly ordered sets as branch spaces of trees satisfying certain natural restrictions designed to make them less complicated than the set are seeking to represent. The authors show that any uncountable order-complete linearly ordered set \(X\) can be represented as the branch space of a tree \(T\) that is more simple than \(X\) and that if \(X\) is representable as a branch space, then any \(G_\delta\)-subset of \(X\) is also representable as a branch space. However, they show that even though the usual set \(\mathbb{R}\) of real numbers can be represented as the branch space of a tree with countable height and countable levels, most subsets of \(\mathbb{R}\) cannot represented in this way. The authors characterize those limit ordinals \(\lambda\) that can be represented by trees whose nodes do not contain …
reviewed by: Dimitru Buşneag
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 54F05
zbMATH DE Number: 2174915
zbMATH Keywords: linear order
zbMATH Keywords: ordinals
zbMATH Keywords: Aronszajn tree
zbMATH Keywords: Aronszajn line
zbMATH Keywords: paracompact
zbMATH Keywords: Lindelöf-perfect space
MaRDI profile type: Publication
full work available at URL: j.topol.2003.12.020
OpenAlex ID: W2086352450
cites work: Lines, trees, and branch spaces
cites work: The Souslin problem
cites work: Q3994935
cites work: Paracompactness in ordered spaces
cites work: Q5541833
cites work: Q5636033
cites work: Q5620774
cites work: A non-metrizable compact linearly ordered topological space, every subspace of which has a 𝜎-minimal base
cites work: Q3221027
cites work: First Countable Spaces that Have Special Pseudo-Bases
DOI: 10.1016/J.TOPOL.2003.12.020
description: scientific article; zbMATH DE number 2174915