Some aspects of the theory of dimension of completely regular spaces
type: Item
instance of: scholarly article
title: Some aspects of the theory of dimension of completely regular spaces
zbMATH Open document ID: 1055.54015
author: I. Tsereteli
author: l. Zambakhidze
published in: Topology and its Applications
publication date: 2004-10-01T00:00:00Z
review text: Let \(\mathcal S\) denote the class of separable metrizable spaces. For any \(X\) from \(\mathcal S\), the three dimension functions, dim, Ind and ind have a common value, say \(d(X)\), which lies in \(N'=\{\dots,-1,0,1,2,\dots\}\cup\{\infty\}\) [see \textit{W. Hurewicz} and \textit{H. Wallman}, ``Dimension theory'' (1941; Zbl 0060.39808)]. It is also true that d satisfies certain properties (e.g., the logarithmic and additive properties). The authors list eight such properties and then three questions one might ask about such a dimension function enjoying some, or all, of them. In this paper, it is the third question, denoted (\(\gamma\)), that they consider, so let us state it here. Question: Do there exist \(N'\)-valued topologically invariant functions defined on classes of spaces wider than the class \(\mathcal S\) and having the above-mentioned eight properties simultaneously? In particular, does a function \(d\) of that type exist on the class of all Tychonoff spaces? If not, then what can one say about the existence of \(N'\)-valued topologically invariant functions defined on the given wider class of spaces and having simultaneously the properties from any preassigned subsystem of these eight? The aim of the paper is to consider this Question for the class \(\mathcal T\) of all completely regular spaces. The authors list eight different classes of functions which correspond respectively to the eight properties that such a dimension function \(d\) might have. The problem then is to select some (possibly proper) subset of these eight classes and then to ask if there is a function \(d\) common to these classes, i.e., to ask if this is ``realizable''. Such a function \(d\) is (awkwardly) called a ``realizator''. Of course for the class \(\mathcal S\), everything works. But the question of whether this is true with the expanded universe \(\mathcal T\) of completely regular spaces can have different answers. The third section of the paper is dedicated to the nonr…
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 2104471
zbMATH Keywords: dimension function
zbMATH Keywords: Martin's axiom
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
full work available at URL: j.topol.2003.09.011
OpenAlex ID: W2033091742
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DOI: 10.1016/J.TOPOL.2003.09.011
description: scientific article; zbMATH DE number 2104471