Characterizing certain incomplete infinite-dimensional absolute retracts
type: Item
instance of: scholarly article
title: Characterizing certain incomplete infinite-dimensional absolute retracts
zbMATH Open document ID: 0629.54011
author: Mladen Bestvina
author: Jerzy Mogilski
published in: Michigan Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: Let \(\sigma =\{t\in [-1,1]^{\infty}| t_ i=0\) for all but finitely many \(i\}\) and let \(\Sigma =\{t\in Q^{\infty}| t_ i=0\) for all but finitely many \(i\}\) where \(Q=[-1,1]^{\infty}\). The second author has obtained characterization theorems for \(\sigma\) and \(\Sigma\) [Proc. Am. Math. Soc. 92, 111-118 (1984; Zbl 0577.57005)]. The authors obtain characterizations for \(\sigma\) and \(\Sigma\) as corollaries of a more general characterization theorem for certain incomplete subsets of Hilbert spaces. This more general characterization theorem also yields a characterization of \(\Sigma\times \sigma\) and gives characterizations of certain absolute Borel sets and of certain ARs. A triangulation theorem for the spaces characterized is also obtained. One version of a general characterization theorem obtained is as follows. If \({\mathcal C}\) is an additive topological class hereditary with respect to closed subsets, and if there exists a \({\mathcal C}\) absorbing set \(\Omega\) in a Hilbert space, then an AR X is homeomorphic to \(\Omega\) if and only if X is \({\mathcal C}_{\sigma}\), X is strongly \({\mathcal C}\) universal and \(X=\cup^{\infty}_{i=1}X_ i\) where each \(X_ i\) is a strong Z-set in X. X is strongly \({\mathcal C}\) universal if for each map f from C to X, where \(C\in {\mathcal C}\), for each closed subset D of C such that \(f|_ D\) is a Z-embedding, and for each open cover \({\mathcal U}\) of X, there exists a Z-embedding h from C into X such that \(h|_ D=f|_ D\) and such that f and h are \({\mathcal U}\) close.
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 54F65
Mathematics Subject Classification ID: 54C50
zbMATH DE Number: 4024272
zbMATH Keywords: strong discrete approximation property
zbMATH Keywords: strong universality
zbMATH Keywords: c-absorbing sets in s-manifolds
zbMATH Keywords: incomplete ANRs
zbMATH Keywords: absolute Borel sets
zbMATH Keywords: triangulation theorem
zbMATH Keywords: strong Z-set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003410
OpenAlex ID: W2034792117
DOI: 10.1307/MMJ/1029003410
description: scientific article; zbMATH DE number 4024272