Density properties and points of uncountable order for families of open sets in function spaces
type: Item
instance of: scholarly article
title: Density properties and points of uncountable order for families of open sets in function spaces
zbMATH Open document ID: 1019.54005
DOI: 10.1016/S0166-8641(01)00157-2
author: Oleg Okunev
author: Vladimir Tkachuk
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Let \(X\) be a Tychonoff space and \(C_p(X)\) denote the space of all continuous real valued functions on \(X\) with the topology of pointwise convergence. The aim of the present work is to study the relation between various cardinal functions on \(X\) and those on \(C_p(X)\). For each subset \(A\subset X\) and each cardinal \(\kappa\) let \([A]_\kappa= \cup \{\overline B:B \subset A\) and \(|B|\leq \kappa\}\). We define the tightness \(t(X)= \min\{\kappa: \overline A=[A]_\kappa\) for all \(A\subset X\}\). We say that \(A\) is \(\kappa\)-dense in \(X\) in case \([A]_\kappa=X\). We define the density tightness \(dt(X)=\min \{\kappa\): every dense subset of \(X\) is \(\kappa\)-dense in \(X\}\). The authors show that \(dt(C_p(X))= t(C_p(X))\) for any \(X\). For an infinite cardinal \(\mu\) we say that \(X\) is \((\kappa,\mu)\)-compact in case, for every subset \(\{x_\alpha: \alpha\in A\} \subset X\) with \(|A|=\kappa\), there is an accumulation point \(x\in X\) such that, if \(U\) is a neighborhood of \(x\), then \(|\{\alpha: x_\alpha\in U\} |\geq\mu\). Among their other results is a characterization of the \((\kappa,\mu)\)-compactness of finite powers of \(X\) in terms of the topology of \(C_p(X)\). A number of unsolved problems are also posed.
Mathematics Subject Classification ID: 54A25
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54D25
zbMATH DE Number: 1787573
zbMATH Keywords: \(\pi\)-characters
zbMATH Keywords: cardinal functions
zbMATH Keywords: tightness
zbMATH Keywords: density tightness
reviewed by: James V. Whittaker
MaRDI profile type: Publication
cites work: Q4228099
cites work: Q4150804
cites work: Some properties of C(X). I
cites work: A method for constructing examples of M-equivalent spaces
cites work: Weak topology of an associated space and \(t\)-equivalence
cites work: Homeomorphisms of function spaces and hereditary cardinal invariants
cites work: Q4064169
full work available at URL: s0166-8641(01)00157-2
OpenAlex ID: W2048988959
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description: scientific article; zbMATH DE number 1787573