The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic
type: Item
instance of: scholarly article
title: The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic
zbMATH Open document ID: 0801.54005
author: Tadeusz Dobrowolski
author: Leonard R. Rubin
published in: Pacific Journal of Mathematics
publication date: 1994-11-29T00:00:00Z
review text: The author's summary: ``A notion of true dimension theory is defined to which is assigned a dimension function \(D\). We consider those \(D\) which have an enhanced Bockstein basis; these include \(D=\dim\) and \(D=\dim_ G\) for any abelian group \(G\). We prove that for each countable polyhedron \(K\), the set of compacta \(X\in 2^ Q\) with \(K\in AE(\{X\})\) is a \(G_ \delta\)-subspace. We apply this fact to show that the hyperspace of the Hilbert cube \(Q\) consisting of compacta (or continua) \(X\) with \(D(X)\leq n\) is a \(G_ \delta\)-subspace. Let \(D^{\geq n}\) (resp., \(D^{\geq n}\cap C(Q))\) denote the space of compacta \(X\) (resp., continua) with \(D(X)\geq n\). We prove that \(\{D^{\geq n}\}^ \infty_{n =1}\) and \(\{D^{\geq n} \cap C(Q) \}^ \infty_{n=2}\) are absorbing sequences for \(\sigma\)-compact spaces. This yields that each \(D^{\geq n}\) and \(D^{\geq n+1} \cap C(Q)\) \((n\geq 1)\) is homeomorphic to the pseudoboundary \(B\) of \(Q\); their respective complements are homeomorphic to the pseudointerior of \(Q\); and the intersections \(\bigcap_ nD^{\geq n}\), \(\bigcap_ nD^{\geq n}\cap C(Q)\) are homeomorphic to \(B^ \infty\), the absorbing set for the class of \(F_{\sigma \delta}\)- sets. Results for the hyperspaces of compacta \(X\) for which \(D(X) \geq n\) uniformly are also obtained''.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 589808
zbMATH Keywords: infinite-dimensional compacta
zbMATH Keywords: covering dimension
zbMATH Keywords: cohomological dimension
zbMATH Keywords: hyperspace of the Hilbert cube
MaRDI profile type: Publication
full work available at URL: pjm.1994.164.15
OpenAlex ID: W2043302607
DOI: 10.2140/PJM.1994.164.15
description: scientific article; zbMATH DE number 589808