On the metrizability number and related invariants of spaces. II
type: Item
instance of: scholarly article
title: On the metrizability number and related invariants of spaces. II
zbMATH Open document ID: 0864.54001
DOI: 10.1016/0166-8641(95)00082-8
author: Yanyan Li
published in: Topology and its Applications
publication date: 1996-11-27T00:00:00Z
review text: This paper continues a study begun by these authors in [ibid. 63, 69-77 (1995; Zbl 0860.54005)]. The metrizability number, \(m(X)\), of a space \(X\) is the smallest cardinal \(\kappa\) such that \(X\) can be represented as a union of \(\kappa\) metrizable subspaces. The first countability number, \(fc(X)\), is defined similarly. In this paper a number of conditions on a space \(X\), involving the weight of \(X\) (denoted \(w(X)\)) and \(fc(X)\), are shown to be equivalent to the singular cardinal hypothesis (SCH): if \(2^{\text{cof} (\kappa)}<\kappa\), then \(k^{\text{cof}(\kappa)}=\kappa^+\). For example, SCH is equivalent to the statement ``if \(X\) is a space such that \(|X|> 2^\omega\), \(cf(w(X))>\omega\), and \(w(X)\leq |X|\), then \(|X|=w(X) \cdot fc(X)\).'' The authors also show that if \(X\) is a compact LOTS and \(m(X)\leq \omega\), then \(X\) is metrizable, and for any compact LOTS \(X\), if \(m(X)>\omega\), then \(w(X) \leq m(X)\). Many other results and examples are given.
reviewed by: Jerry E. Vaughan
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 937172
zbMATH Keywords: metrizability number
zbMATH Keywords: first countability number
zbMATH Keywords: singular cardinal hypothesis
zbMATH Keywords: LOTS
MaRDI profile type: Publication
full work available at URL: 0166-8641(95)00082-8
OpenAlex ID: W2015696538
cites work: Q3994935
cites work: Paracompactness in ordered spaces
cites work: Compact spaces representable as unions of nice subspaces
cites work: On the metrizability number and related invariants of spaces
cites work: A Topological Equivalence of the Singular Cardinals Hypothesis
cites work: Q3852172
cites work: Q5636033
cites work: Some set-theoretic constructions in topology
cites work: Q3221027
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description: scientific article; zbMATH DE number 937172