Products of compactifications
type: Item
instance of: scholarly article
title: Products of compactifications
zbMATH Open document ID: 1006.54034
DOI: 10.1023/A:1015672419089
author: E. G. Sklyarenko
author: Gülhan Aslım
author: O. B. Özbakir
published in: Acta Mathematica Hungarica
publication date: 2003-02-18T00:00:00Z
review text: Let \(X\) be a completely regular (Hausdorff) topological space and \(b_1X\), \(b_2X\) two compactifications of \(X\). The natural inclusions \(i_k\:X \to b_kX\), \(k = 1,2\) define a new inclusion \(i\:X \to b_1X \times b_2X\) by \(i(x) = (i_1(x), i_2(x))\) for \(x\in X\). The closure of \(i(X)\) in \(b_1X \times b_2X\) is some new compactification \((b_1\times b_2)X\) of \(X\). The authors call it the product of the compactifications \(b_1X\) and \(b_2X\) of the space \(X\). They study the minimality of the compactification in the family of all compactifications of \(X\) following the previous ones (with respect to an order relation \(\leq\) by continuous surjections preserving \(X\)), the associativity property and several properties. The product \((\prod_\lambda b_\lambda)X\) of finite or infinite compactifications of \(X\) can also be considered in a similar way. The authors show that \(\varprojlim(b_{\lambda_1}\times\dots\times b_{\lambda_k})X = (\prod_\lambda b_\lambda)X\), here \(\{b_\lambda X\}\) is an infinite collection of compactifications of \(X\) and the inverse system \(\{(b_{\lambda_1}\times\dots\times b_{\lambda_k})X\}\) is naturally defined by the collection. Then an application of the theorem to a special compactification of a locally compact space is considered.
reviewed by: Katsuya Yokoi
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 55M10
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 1870470
zbMATH Keywords: direct product
zbMATH Keywords: compactification product
zbMATH Keywords: supplements
zbMATH Keywords: dimension
MaRDI profile type: Publication
full work available at URL: a:1015672419089
OpenAlex ID: W1825775275
description: scientific article; zbMATH DE number 1870470