Cohomological dimension of Tychonov spaces
type: Item
instance of: scholarly article
title: Cohomological dimension of Tychonov spaces
zbMATH Open document ID: 0888.55001
DOI: 10.1016/S0166-8641(96)00176-9
published in: Topology and its Applications
publication date: 1998-05-25T00:00:00Z
review text: The concepts of dimension, including both covering (dim) and cohomological (dim\(_G\) for an abelian group \(G\)), have typically been studied for spaces which are at least normal, although the richest theory is usually developed for metrizable or at least paracompact spaces. A good source for dimension theory is [\textit{R. Engelking}, Theory of dimensions, finite and infinite (1995; Zbl 0872.54002)]; for cohomological dimension, one may consult [\textit{V. I. Kuzminov}, Russ. Math. Surv. 23, No. 5, 1-45 (1968); translation from Usp. Mat. Nauk 23, No. 5(143), 3-49 (1968; Zbl 0179.27903)]. The aim of the current paper is to extend the notion of cohomological dimension to the class of Tikhonov spaces which are not necessarily normal. The author points out that under the usual definition, dim\(_G X<\infty\) implies that \(X\) is normal. The definition chosen by this author agrees with the classical one for normal spaces, but does not force \(X\) to be normal. It is given in terms of extension of maps to \(K(G,n)\) as before, but in a certain different sense. The groups \(G\) considered must be countable, and \(K(G,n)\) is always taken as a Polish ANR-space homotopy equivalent to a corresponding Eilenberg-MacLane complex. One of the first results shows that dim\(_G X=\)dim\(_G\nu X\) where \(\nu X\) is the Hewitt realcompactification of \(X\). Some items proved are: (1) dim\(_\mathbb Z(A\cup B)\leq\) dim\(_\mathbb Z A+\) dim\(_\mathbb Z B+1\) (addition theorem, proved for metrizable spaces in [\textit{L. R. Rubin}, Topology Appl. 40, No. 3, 233-263 (1991; Zbl 0773.55002)]; (2) the existence of universal spaces of a given cohomological dimension and a given weight; (3) limit theorems; (4) a subspace theorem.
Mathematics Subject Classification ID: 55M10
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1067518
zbMATH Keywords: cohomological dimension
zbMATH Keywords: inverse spectrum
zbMATH Keywords: absolute extensor
zbMATH Keywords: Tikhonov space
zbMATH Keywords: Eilenberg-MacLane complex
zbMATH Keywords: addition theorem
zbMATH Keywords: limit theorem
zbMATH Keywords: subspace theorem
zbMATH Keywords: realcompact
zbMATH Keywords: Hewitt realcompactification
zbMATH Keywords: Polish space
zbMATH Keywords: universal space
author: Alex Chigogidze
reviewed by: Leonard R. Rubin
MaRDI profile type: Publication
cites work: Characterization of Hilbert space manifolds revisited
cites work: Spaces in Which Special Sets are z-Embedded
cites work: Q3952791
cites work: On the dimension of increments of Tychonoff spaces
cites work: Q3027895
cites work: A characterization of the dimension dim ofR-compacta, and universal spaces of given weight and dimension
cites work: NONCOMPACT ABSOLUTE EXTENSORS IN DIMENSIONn,n-SOFT MAPPINGS, AND THEIR APPLICATIONS
cites work: \(UV^ n\)-equivalence and \(n\)-equivalence
cites work: Q3996849
cites work: UNIVERSAL MENGER COMPACTA AND UNIVERSAL MAPPINGS
cites work: Q3833251
cites work: THE EILENBERG-BORSUK THEOREM FOR MAPPINGS INTO AN ARBITRARY COMPLEX
cites work: Q3841770
cites work: Q4326502
cites work: Cohomological Dimension and Metrizable Spaces
cites work: Extension theory: The interface between set-theoretic and algebraic topology
cites work: Universal Cell-Like Maps
cites work: Spaces Without Cohomological Dimension Preserving Compactifications
cites work: Q3270120
cites work: HOMOLOGICAL DIMENSION THEORY
cites work: Factorization theorems for cohomological dimension
cites work: Completion Theorem for Cohomological Dimensions
cites work: Universal separable metrizable spaces of given cohomological dimension
cites work: Characterizing cohomological dimension: The cohomological dimension of \(A \cup{} B\)
cites work: Functors and uncountable powers of compacta
cites work: Q5522742
cites work: Q3928675
description: scientific article; zbMATH DE number 1067518