Reflection theorems for cardinal functions
type: Item
instance of: scholarly article
title: Reflection theorems for cardinal functions
zbMATH Open document ID: 0943.54003
DOI: 10.1016/S0166-8641(99)00056-5
author: Richard E. Hodel
author: Jerry E. Vaughan
published in: Topology and its Applications
publication date: 2000-09-04T00:00:00Z
review text: A reflection theorem is a statement of the form ``every structure with a certain property has a small substructure with the same property''. Normally `smaller' means `of cardinality \(\kappa\) or less' for certain \(\kappa\). In this paper a cardinal function \(\varphi\) is said to (strongly) reflect \(\kappa\) if every space \(X\) with \(\varphi(X)\geq\kappa\) has a subspace \(Y\) with \(|Y|\leq\kappa\) and \(\varphi(Y)\geq\kappa\) (\(\varphi(Z)\geq\kappa\) whenever \(Y\subseteq Z\subseteq X\)). A model result is \textit{A. Hajnal} and \textit{I. Juhász}'s theorem [Proc. Am. Math. Soc. 79, 657-658 (1980; Zbl 0432.54003)] that weight strongly reflects every infinite cardinal. The authors prove reflection theorems for many familiar cardinal functions, e.g., \(c\), \(s\), \(e\), \(hd\) and \(hL\) strongly reflect all infinite cardinals; and \(t\) and \(\chi\) reflect all cardinals in the class of compact Hausdorff spaces. The final section contains a generalization of \textit{A. Dow}'s metrization theorem [ibid. 104, No. 3, 999-1001 (1988; Zbl 0692.54018)] to higher cardinals: if \(X\) is initially \(\kappa\)-compact, \(\chi(X)\leq\kappa\) and \(w(X)\geq\kappa^+\) then there is \(Y\subseteq X\) of cardinality at most \(\kappa^+\) such that \(pw(Y)\geq\kappa^+\), where \(pw\) is the point-weight, the minimum order of a base for the space.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 1395072
zbMATH Keywords: cardinal function
zbMATH Keywords: compactness
zbMATH Keywords: weight
zbMATH Keywords: point-weight
MaRDI profile type: Publication
full work available at URL: s0166-8641(99)00056-5
OpenAlex ID: W1983664821
description: scientific article; zbMATH DE number 1395072