On the finite-dimensionality of topological products
type: Item
instance of: scholarly article
title: On the finite-dimensionality of topological products
zbMATH Open document ID: 0896.54020
DOI: 10.1016/S0166-8641(98)00064-9
author: B. A. Pasynkov
published in: Topology and its Applications
publication date: 1998-09-21T00:00:00Z
review text: This is an important paper in (at least) dimension theory. In this paper a complete and well-prepared proof (i.e. we believe that any standard textbook concerning dimension theory can accept it as a wonderful proof of this beautiful theorem) is given for one of his well-known announced results in [Sov. Math., Dokl. 10, 1402-1406 (1969); translation from Dokl. Akad. Nauk SSSR 189, 254-257 (1969; Zbl 0205.52803)]. In particular, a detailed proof of the following (nearly 30 years old) beautiful theorem is given here for the first time: If \(X\) is paracompact, \(Y\) is locally compact and \(\text{Ind }X, \text{Ind }Y <\infty\), then \(\text{Ind }X\times Y <\infty\). Moreover, there exist integers \(d(k,l)\leq -1\) for \(k,l = -1,0,1,\dots\) such that \(\text{Ind } X\times Y\geq d(\text{Ind } X, \text{Ind }Y)\). Moreover, the author gives here a proof for the more general case of (not necessarily paracompact) normal product space \(X\times Y\) in Theorem 1.1. Let the reviewer remind the reader that there still remain lots of good results due to the author, whose proofs have not been published yet (e.g. the reader can see many of them in a survey article [\textit{B. A. Pasynkov, V. V. Fedorchuk} and \textit{V. V. Filippov}, J. Sov. Math. 18, 789-841 (1982); translation from Itogi Nauki Tekh., Ser. Algebra Topologiya Geom. 17, 229-306 (1979; Zbl 0444.54024)]). Let me expect the author, one of the authorities in dimension theory, to publish detailed proofs for them, since no one (as far as I know) has reproved them over 20 years!
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 54B10
zbMATH DE Number: 1121474
zbMATH Keywords: large inductive dimension
reviewed by: Koichi Tsuda
MaRDI profile type: Publication
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cites work: Q3977514
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cites work: On Subset Theorems and the Dimension of Products
cites work: On the Dimension of Product Spaces
cites work: On the dimension of the product of Tychonoff spaces
cites work: Extensions of zero-sets in the product of topological spaces
cites work: Dimension of metric spaces and Hilbert’s problem 13
cites work: Об универсальных бикомпактных и метрических пространствах данной размерности
cites work: Q5605712
cites work: Q4101156
cites work: Factorization theorems in dimension theory
cites work: Products with a metric factor
cites work: Q3932058
cites work: An n-dimensional version of Wage's example
full work available at URL: s0166-8641(98)00064-9
OpenAlex ID: W1970755222
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description: scientific article; zbMATH DE number 1121474