Kurepa trees and topological non-reflection
type: Item
instance of: scholarly article
title: Kurepa trees and topological non-reflection
zbMATH Open document ID: 1076.54003
author: Piotr Koszmider
published in: Topology and its Applications
publication date: 2005-06-10T00:00:00Z
review text: The author is mainly interested in a Kurepa tree. A tree is called a Kurepa tree if it has countable levels and more than \(\omega_1\) uncountable branches. The existence of a Kurepa tree is called Kurepa's hypothesis (KH). All trees are assumed to be of height \(\omega_1\). \(T\) is called a tree with many branches if for every \(t\) in \(T\), there is an uncountable branch of \(T\) whose range contains \(t\). \(\text{Br}(T)\) denotes the collection of all uncountable branches of \(T\). KH\(^\lambda(T)\) denotes the following statement: There is an \(\omega_1\)-tree \(T\) with exactly \(\lambda\)-many branches and \(\lambda^\omega= \lambda\). The main object investigated in this paper is \({\mathcal F}(T)\) -- some well-founded and cofinal subfamily of \([\text{Br}(T)]^\omega\), the family of all countable collections of branches of \(T\), for any normal tree \(T\) with many branches. The topology on \({\mathcal F}(T)\) is defined by \[ {\mathcal B}(T)= \{U(\theta, F): F\in [\text{Br}(T)]^\omega, \theta< \omega_1\}\cup\{\phi\} \] as its basis, where for \(F\in[\text{Br}(T)]^{<\omega}\) and \(\theta<\omega_1\), \[ {\mathcal F}_\theta(T)= \{X\in [\text{Br}(T)]^\omega:\forall t\in \text{Lev}_\theta T\exists ! b\in X,\;t\in\text{ran}(b)\} \] and \[ U(\theta, F)= \Biggl\{X\in\bigcup_{0\leq\theta_1\leq\theta} {\mathcal F}_{\theta_1}(T): F\subset X\Biggr\}. \] The author proves the following theorems as main results: Theorem 35. Suppose KH\(^\lambda(T)\) holds for \(\lambda>\omega_1\). Then there is a zero-dimensional, regular (weakly collectionwise Hausdorff, if \(\text{cf}(\lambda)>\omega_1\)) locally metrizable, non-\(\sigma\)-para-Lindelöf, hence, nonparacompact, nonmetrizable space \({\mathcal F}(T)\) of size \(\lambda\) such that every one of its subspaces of size less than \(\lambda\) is metrizable and included in a clopen metrizable subspace. Moreover, \({\mathcal F}(T)\) is not a generalized ordered space nor a monotonically normal space. \(S(T)\) denotes the sta…
reviewed by: Akihiro Okuyama
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 03E04
Mathematics Subject Classification ID: 54D15
Mathematics Subject Classification ID: 54D20
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 2174906
zbMATH Keywords: Kurepa tree
zbMATH Keywords: KH\(^\lambda(T)\), \(S(T)\)
zbMATH Keywords: topological non-reflection
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.033
OpenAlex ID: W2058264183
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DOI: 10.1016/J.TOPOL.2003.08.033
description: scientific article; zbMATH DE number 2174906