The behaviour of dimension functions on unions of closed subsets
type: Item
instance of: scholarly article
title: The behaviour of dimension functions on unions of closed subsets
zbMATH Open document ID: 1063.54026
author: Vitalij A. Chatyrko
author: Yasunao Hattori
author: Michael G. Charalambous
published in: Journal of the Mathematical Society of Japan
publication date: 2004-10-27T00:00:00Z
review text: Let \(d\) be a dimension function. For ordinals \(\beta, \alpha\) such that \(\beta<\alpha\), and a space \(X\) of a class \(\mathcal K\) of topological spaces with \(dX=\alpha\), the authors define the values \(m(X,d,\beta,\alpha)=\min\{k:X=\bigcup_{i=1}^kX_i\), where \(X_i\) is closed in \(X\) and \(dX_i\leq\beta\}\), \(m_{\mathcal K}(d,\beta,\alpha)=\min\{m(X,d,\beta,\alpha): X\in{\mathcal K}\) and \(m(X,d,\beta,\alpha)\) exists\(\}\) and \(M_{\mathcal K}(d,\beta,\alpha)=\sup\{m(X,d,\beta,\alpha): X\in{\mathcal K}\) and \(m(X,d,\beta,\alpha)\) exists\(\}\). In this paper, they determine the values of \(m_{\mathcal P}(\)Cmp,\(\beta,\alpha)\), \(M_{\mathcal P}(\)Cmp,\(\beta,\alpha)\), \(m_{\mathcal C}(\)trInd,\(\beta,\alpha)\) and \(M_{\mathcal C}(\)trInd,\(\beta,\alpha)\), where \({\mathcal P}\) and \({\mathcal C}\) mean the classes of separable completely metrizable spaces and metrizable compact spaces, respectively. Furthermore, they introduce new dimension functions: the additive compactness degree Cmp\(_{\cup}\) and the transfinite additive inductive dimension functions ind\(_{\cup}\) and Ind\(_{\cup}\), and discuss the values \(m_{\mathcal P}(\)Cmp\(_{\cup},\beta,\alpha)\), \(M_{\mathcal P}(\)Cmp\(_{\cup},\beta,\alpha)\), \(m_{\mathcal C}(\)ind\(_{\cup},\beta,\alpha)\), \(m_{\mathcal C}(\)Ind\(_{\cup},\beta,\alpha)\), \(M_{\mathcal C}(\)ind\(_{\cup},\beta,\alpha)\) and \(M_{\mathcal C}(\)Ind\(_{\cup},\beta,\alpha)\).
reviewed by: Kohzo Yamada
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 2110893
zbMATH Keywords: large inductive compactness degree
zbMATH Keywords: transfinite large inductive dimension
zbMATH Keywords: additive compactness degree
zbMATH Keywords: transfinite additive inductive dimension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191418641
OpenAlex ID: W2048073338
DOI: 10.2969/JMSJ/1191418641
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description: scientific article; zbMATH DE number 2110893