Zero-dimensional spaces from linear structures
type: Item
instance of: scholarly article
title: Zero-dimensional spaces from linear structures
zbMATH Open document ID: 1020.54018
DOI: 10.1016/S0019-3577(01)80043-4
author: Jerry E. Vaughan
published in: Indagationes Mathematicae. New Series
publication date: 2003-10-06T00:00:00Z
review text: This paper contains some interesting results concerning topological spaces which are stratifiable over regular infinite cardinals \(\omega_\mu\). These were introduced by the author in [Pac. J. Math. 43, 253-266 (1972; Zbl 0226.54027)]. Example 1. There exists an \(\omega_\mu\)-Nagata space that does not have an ortho-base. In particular, for \(\omega_\mu= \omega_0\), there exists a Nagata space that does not have an ortho-base. Example 2. For every \(\omega_\mu\) there exists an \(\omega_\mu\)-stratifiable space that is not \(\kappa\)-Nagata for any cardinal \(\kappa\). In particular, there exists such a space that is stratifiable. Example 3. There exists a stratifiable space \(X\) such that the box product \(\square X^{\omega_0}\) is stratifiable over \(\omega_1\) (hence hereditarily paracompact and monotonically normal) but not stratifiable. Example 4. For \(\mu> 0\) there exists a space \(X\) which is stratifiable over \(\omega_\mu\) (and no other regular cardinal) such that \(\omega= \psi(X)< \omega_\mu< \omega_{\mu+1}\leq \chi(X)\), where \(\psi(X)\) is the pseudocharacter and \(\chi(X)\) is the character of \(X\). Example 5. For every \(\omega_\mu\) there exists an \(\omega_\mu\)-stratifiable space \(X\) such that \(|X|= \omega_\mu\) and every point of \(X\) has character greater than \(\omega_\mu\). Theorem. Every \(\omega_1\)-additive paracompact space is ultraparacompact. In particular, every \(\omega_1\)-additive linearly stratifiable space is ultraparacompact.
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D70
Mathematics Subject Classification ID: 54G20
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D20
zbMATH DE Number: 1893622
zbMATH Keywords: stratifiable space
zbMATH Keywords: \(\omega_1\)-additive space
zbMATH Keywords: \(\omega_\mu\)-Nagata space
zbMATH Keywords: ortho-base
zbMATH Keywords: box product
zbMATH Keywords: pseudocharacter
zbMATH Keywords: character
zbMATH Keywords: paracompact
zbMATH Keywords: ultraparacompact
zbMATH Keywords: linearly stratifiable space
reviewed by: Harald Brandenburg
MaRDI profile type: Publication
cites work: On stratifiable spaces
cites work: Some generalizations of metric spaces
cites work: The box product of countably many metrizable spaces need not be normal
cites work: Another nonnormal box product
cites work: Mapping Theorems for Non-Compact Spaces
cites work: Q5526585
cites work: Extending continuous functions on zero-dimensional spaces
cites work: Q3994935
cites work: Q4185470
cites work: On Stratifiable and Elastic Spaces
cites work: Q5669955
cites work: A relation between perfect separability, completeness, and normality in semi-metric spaces
cites work: Yet Another Note on Paracompact Spaces
cites work: Q5796855
cites work: Q5630369
cites work: Q4104980
cites work: Q4074084
cites work: Nonequality of Dimensions for Metric Spaces
cites work: Remarks on some topological spaces of high power
cites work: Linearly stratifiable spaces
cites work: Linearly Ordered Collections and Paracompactness
Recommended article: Set-theoretic constructions of nonshrinking open covers
Recommended article: Characterizations of expandability and the \({\mathcal B}\)-property
Recommended article: Q5112608
Recommended article: Q4430826
Recommended article: Q4326518
Recommended article: Q4314646
Recommended article: Some results on \([n,m]\)-paracompact and \([n,m]\)-compact spaces
Recommended article: \(M_ 3\)-spaces whose every point has a closure preserving outer base are \(M_ 1\)
Recommended article: Q4809202
Recommended article: On functions and stratifiable \(\mu\)-spaces
description: scientific article; zbMATH DE number 1893622