Calibers, free sequences and density
type: Item
instance of: scholarly article
title: Calibers, free sequences and density
zbMATH Open document ID: 1014.54005
DOI: 10.1016/S0166-8641(01)00071-2
published in: Topology and its Applications
publication date: 2002-06-24T00:00:00Z
review text: This paper provides some interesting results concerning the influence of several notions of caliber on the density of a space that is a union of a ``small'' number of compact sets which have no ``long'' free sequences. A cardinal \(\kappa\) is said to be a caliber of a space \(X\) (\(\kappa\in Cal(X)\)) provided that among any family of \(\kappa\) many open sets there is a subfamily of cardinality \(\kappa\) that has non-empty intersection. Obviously if the cofinality of \(\kappa\) is larger than the density of \(X\) (\(cf(\kappa) > d(X)\)), then \(\kappa\in Cal(X)\), but the converse in not true in general. This paper is concerned with conditions which will imply the converse, or at least provide an upper bound for \(d(X)\). A sequence \(\{x_\alpha:\alpha<\kappa\}\) is called a free sequence (of length \(\kappa\)) provided that for all \(\gamma <\kappa\), \(cl_X(\{x_\alpha:\alpha<\gamma\})\cap cl_X( \{x_\alpha:\gamma\leq \alpha<\kappa\})=\emptyset\) (the concept is due to A. Arhangel\('\)skii). The authors prove: if \(X\) is a \(T_3\)-space, \(\kappa\in Cal(X)\), and \(X\) is the union of at most \(\kappa\) many compact subsets each with no free sequences of length \(\kappa\), then \(d(X) <\kappa\). Since countable tightness implies there are no uncountable free sequences, a new corollary to the previous result states: If \(\omega_1\) is a caliber of \(X\) and \(X\) is the union of \(\omega_1\) compact sets each of countable tightness, then \(X\) is separable. The authors say that \((\lambda,\kappa)\) is a (pair) caliber of \(X\) (\((\lambda,\kappa)\in Cal_2(X)\)) if \(\lambda\leq \kappa\) and among any family of \(\kappa\) many open sets there is a subfamily of cardinality \(\lambda\) with non-empty intersection. The authors prove: If \(X\) is a \(T_3\)-space and \((\mu,\kappa)\) is a caliber of \(X\), and \(X\) is the union of at most \(\kappa\) many compact sets each having no free sequences of length \(\mu\), then \(d(X) <\kappa\). The authors say that \((\mu,\…
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54D65
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 1758568
zbMATH Keywords: caliber
zbMATH Keywords: free sequences
zbMATH Keywords: density
zbMATH Keywords: tightness
zbMATH Keywords: pair caliber
zbMATH Keywords: triple caliber
author: István Juhász
author: Zoltán Szentmiklóssy
reviewed by: Jerry E. Vaughan
MaRDI profile type: Publication
cites work: Projective \(\sigma\)-compactness, \(\omega_1\)-caliber, and \(C_p\)-spaces
cites work: Q3938195
cites work: Q3471188
full work available at URL: s0166-8641(01)00071-2
OpenAlex ID: W2009586930
description: scientific article; zbMATH DE number 1758568