Computation of inductive dimensions of product of compacta
type: Item
instance of: scholarly article
title: Computation of inductive dimensions of product of compacta
zbMATH Open document ID: 1300.54048
author: A. Karassev
author: K. L. Kozlov
published in: Topology and its Applications
publication date: 2014-10-24T00:00:00Z
reviewed by: Dimitris Georgiou
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 6361146
zbMATH Keywords: large (small) inductive dimension
zbMATH Keywords: normal base
zbMATH Keywords: base dimension \(I\)
MaRDI profile type: Publication
full work available at URL: j.topol.2014.08.023
OpenAlex ID: W1992673074
cites work: Q5180089
cites work: The dimension of metrizable subspaces of Eberlein compacta and Eberlein compactifications of metrizable spaces
cites work: Some estimates of the inductive dimensions of the union of two sets
cites work: Q3012600
cites work: Q4457313
cites work: Q4150804
cites work: Q4855460
cites work: Q5654759
cites work: The inductive dimension of a space by its normal base
cites work: The covering dimension invariants
cites work: Q4654006
cites work: On the dimension of partial products
cites work: Q5581269
cites work: Q5605712
cites work: On the finite-dimensionality of topological products
DOI: 10.1016/J.TOPOL.2014.08.023
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review text: There are several reasons to study dimension-like invariants: to get more information about spaces; to extend the class of spaces under investigation; to evaluate the classical dimensions, etc. M. Charalambous, A. Chigogidze, V. Filippov, J. Aarts, T. Nishiura, B. Pasynkov, S. Iliadis are a few to be mentioned in connection with these themes of research. The last works of V. V. Fedorchuk are connected with the problem of finding unifying approaches to the study of set-theoretical methods in the classical dimension theory and algebraic methods in the investigation of cohomological dimensions. In this paper the authors demonstrate how base dimension \(I\) can be used to compute the classical dimension Ind of products. \textit{B. A. Pasynkov} [Sov. Math., Dokl. 10, 1402--1406 (1969); translation from Dokl. Akad. Nauk SSSR 189, 254-257 (1969; Zbl 0205.52803)] established the finite-dimensionality of the product of two finite-dimensional compacta. \textit{V. V. Filippov} [Sov. Math., Dokl. 13, 250--254 (1972); translation from Dokl. Akad. Nauk SSSR 202, 1016-1019 (1972; Zbl 0243.54032)] constructed an example of compact spaces \(X\), \(Y\) such that NEWLINE\[NEWLINE{\mathrm{ Ind}}X = 1,\;{\mathrm{ Ind}}Y = 2\;{\mathrm{ and}} \;{\mathrm{ Ind}} X\times Y \geq {\mathrm{ ind} } X\times Y>3.NEWLINE\]NEWLINE By Pasynkov's estimation \({\mathrm{ Ind}}X \times Y\leq 7\), and \({\mathrm{ ind}}X \times Y\leq 4\), see \textit{V. A. Chatyrko} and \textit{K. L. Kozlov} [Commentat. Math. Univ. Carol. 41, No. 3, 597--603 (2000; Zbl 1038.54012)].NEWLINENEWLINEIn this very interesting paper the authors give the following result:NEWLINENEWLINE{ Theorem.} For compact spaces \(X\), \(Y\) in Filippov's example we have NEWLINE\[NEWLINE{\mathrm{ Ind}}X = 1, \;{\mathrm{ Ind}} Y = 2\;{\mathrm{ and}}\;{\mathrm{ Ind}} X \times Y = {\mathrm{ ind}}X \times Y = 4.NEWLINE\]
description: scientific article; zbMATH DE number 6361146