On \({\mathbb{R}}^{\infty}\)-manifolds and \(Q^{\infty}\)-manifolds. II: Infinite deficiency
type: Item
instance of: scholarly article
title: On \({\mathbb{R}}^{\infty}\)-manifolds and \(Q^{\infty}\)-manifolds. II: Infinite deficiency
zbMATH Open document ID: 0576.57016
author: K. Sakai
published in: Tsukuba Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: This paper is a sequel to Topology Appl. 18, 69-79 (1984; Zbl 0568.57014). In this paper, as generalizations of closed sets contained in collared subsets, we introduce the notion of D-sets \((D^*\)-sets) and establish the unknotting theorem for D-sets in a manifold modelled on \({\mathbb{R}}^{\infty}=dir \lim {\mathbb{R}}^ n\) or \(Q^{\infty}=dir \lim Q^ n\), where Q is the Hilbert cube. This yields equality of D-sets, \(D^*\)- sets and infinite (i.e., \({\mathbb{R}}^{\infty}\)- or \(Q^{\infty}\)-) deficient sets. Our theorem corresponds to a weak version of the unknotting theorem for infinite deficient sets proved by V. T. Liem. However, our proof is elementary and short. And we give an alternative proof of the infinite deficient embedding approximation theorem due to Liem. Using Anderson-McCharen's trick, this approximation theorem strengthens our unknotting theorem in the strong form. Moreover, we show that the union of two \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifolds meeting in an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold is also an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold, and that for any space X, \(X\times {\mathbb{R}}\) is an \({\mathbb{R}}^{\infty}\)- (or \(Q^{\infty}\)-) manifold if and only if so is \(X\times I\). And we have also an alternative proof of the collaring theorem established by Liem. Recently, the author obtained the negative answer of Problem 6-4 in this paper.
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57N37
zbMATH DE Number: 3921542
zbMATH Keywords: \({\mathbb{R}}^{\infty }\)-manifold
zbMATH Keywords: \(Q^{\infty }\)-manifold
zbMATH Keywords: closed sets contained in collared subsets
zbMATH Keywords: unknotting theorem for D-sets
zbMATH Keywords: infinite deficient sets
zbMATH Keywords: infinite deficient embedding approximation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496159948
OpenAlex ID: W1519419456
DOI: 10.21099/TKBJM/1496159948
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description: scientific article; zbMATH DE number 3921542