Characterization of compact spaces with noncoinciding dimensions which are subsets of products of simple spaces
type: Item
instance of: scholarly article
title: Characterization of compact spaces with noncoinciding dimensions which are subsets of products of simple spaces
zbMATH Open document ID: 1152.54024
published in: Topology and its Applications
publication date: 2008-11-12T00:00:00Z
review text: First examples of compact spaces with noncoinciding classical dimensions ind and \(\dim\) were realized as closed subsets of the product of the long closed interval \([0,\omega_c]\), where \(\omega_c\) is the first ordinal of cardinality of the continuum, and the closed bounded interval \([0,1]\). In the present paper the author studies dimensional properties of closed subsets of products \(\Pi=[0,\omega_c] \times K\), where \(K\) is a metrizable compactum. The first result is the following: for any closed subset \(X\) of \(\Pi\) we have \(\operatorname{Ind}X \leq \dim X +1.\) This implies the splitting of all closed subsets \(X\) of \(\Pi\) into two classes: either \(\operatorname{Ind}X= \operatorname{ind}X= \dim X +1\) or \(\dim X= \operatorname{ind}X \leq \operatorname{Ind} X\). The second (main) result is a characterization of closed subsets \(X\) of \(\Pi\) having \(\dim X = 1\) and \(\operatorname{ind}X= \operatorname{Ind}X =2\). Let us note that the existence of a closed subset \(X\) of \(\Pi\) with \(\dim X= \operatorname{ind}X< \operatorname{Ind}X\) is an open problem.
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 5365215
zbMATH Keywords: compactum
zbMATH Keywords: covering dimension
zbMATH Keywords: inductive dimensions
zbMATH Keywords: product
author: K. L. Kozlov
reviewed by: Vitalij A. Chatyrko
MaRDI profile type: Publication
full work available at URL: j.topol.2007.05.031
OpenAlex ID: W1991191264
cites work: Q5180089
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cites work: On an approach to constructing compacta with different dimensions dim and ind
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cites work: The Urysohn identity and dimension of manifolds
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cites work: Q5605712
DOI: 10.1016/J.TOPOL.2007.05.031
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description: scientific article; zbMATH DE number 5365215