Vertical order in the Hilbert cube
type: Item
instance of: scholarly article
title: Vertical order in the Hilbert cube
zbMATH Open document ID: 0623.57008
author: Dennis J. Garity
author: David G. Wright
published in: Illinois Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The paper addresses the problem of detecting ''tameness'' of closed subsets of the Hilbert cube that are situated with ''small'' vertical order (with respect to some factorization \(I^{\infty}\times I)\). The main result is that a compact subset \(A\) that has infinite codimension and has countable vertical order (i.e., the cardinality of \(A\cap (\{q\}\times I)\) is countable for each \(q\in I^{\infty})\) is a \(Z\)-set. The result contrasts with the situation for Euclidean space \(E^n\), \(n\geq 4\), for there are wild subsets of each dimension \(\leq n-1\) that have finite vertical order. The result parallels the situation in \(E^3\) where 1-dimensional compact subsets with countable vertical order are tame.
Mathematics Subject Classification ID: 57N45
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 4011438
zbMATH Keywords: tameness of closed subsets of Hilbert cube
zbMATH Keywords: vertical order
zbMATH Keywords: Z-set
reviewed by: John Walsh
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/ijm/1256069294
OpenAlex ID: W1566453480
DOI: 10.1215/IJM/1256069294
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description: scientific article; zbMATH DE number 4011438