Subsets of \(^{\omega}\omega\) and generalized metric spaces
type: Item
instance of: scholarly article
title: Subsets of \(^{\omega}\omega\) and generalized metric spaces
zbMATH Open document ID: 0555.03022
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Recall the Fréchet order on \(^{\omega}\omega:f<^*g\) iff, for all but finitely many n, \(f(n)<g(n)\). A \(\kappa\)-scale is a well-ordered dominating (under the Fréchet order) family of type \(\kappa\). The statement BF(\(\kappa)\) says that if \(A\subset^{\omega}\omega\) and \(| A| <\kappa\) then there is some g with \(f<^*g\) for all \(f\in A\). CH implies both ''there is an \(\omega_ 1\)-scale'' and BF(\({\mathfrak c})\), where \({\mathfrak c}\) is the cardinality of the continuum. In this paper, the authors derive statements about generalized metric spaces known to follow from CH from statements about the Fréchet order. For example: using Heath's bow-tie space, they get a Lindelöf semimetrizable space with no countable network from an \(\omega_ 1\)- scale; from BF(\(\kappa)\) they get ''every sequential space of character less than \(\kappa\) is Fréchet''; and from BF(\({\mathfrak c})\) they get ''every compact Hausdorff space which is weakly first countable has size either \(\leq \omega\) or \(\geq {\mathfrak c}.''\)
reviewed by: Judith Roitman
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E25
zbMATH DE Number: 3884147
zbMATH Keywords: Fréchet order
zbMATH Keywords: scale
zbMATH Keywords: generalized metric spaces
zbMATH Keywords: CH
author: Dennis Burke
author: Sheldon W. Davis
MaRDI profile type: Publication
full work available at URL: pjm.1984.110.273
OpenAlex ID: W2065859974
DOI: 10.2140/PJM.1984.110.273
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description: scientific article; zbMATH DE number 3884147