A short elementary proof of the Dranishnikov-West theorem on stable intersection of compacta in Euclidean spaces
type: Item
instance of: scholarly article
title: A short elementary proof of the Dranishnikov-West theorem on stable intersection of compacta in Euclidean spaces
zbMATH Open document ID: 0871.54038
DOI: 10.1016/S0166-8641(96)00053-3
author: Yaki Sternfeld
published in: Topology and its Applications
publication date: 1997-10-05T00:00:00Z
review text: In early 1990's A. N. Dranishnikov and J. E. West proved the following important result concerning the stability of intersections of compacta in \(\mathbb{R}^n\): Suppose \(X\) and \(Y\) are arbitrary compacta, satisfying the condition that \(\dim X\times Y=n\). Then there exist maps \(f:X \to\mathbb{R}^n\) and \(g:Y \to\mathbb{R}^n\) and \(\varepsilon>0\) such that for every maps \(f'\) and \(g'\) \(\varepsilon\)-close to \(f\) and \(g\), respectively, \(f'(X)\) and \(g'(Y)\) always intersect (i.e. \(X\) and \(Y\) have a stable intersection) [\textit{A. N. Dranishnikov} and \textit{J. West}, Topology Appl. 43, No. 2, 181-187 (1992; Zbl 0791.54048)]. In the present note the author gives an elegant short and elementary proof of this result, using only two classical results: the Baire theorem and a theorem of Aleksandrov which asserts that for every \(n\)-dimensional compact subset \(Z\subset \mathbb{R}^k\) there exists a \((k-n)\)-dimensional polyhedron \(P\subset \mathbb{R}^k\) such that \(Z\) is not removable from \(P\) [\textit{P. Aleksandrov} (Alexandroff), Nachrichten Göttingen 37, 25-44 (1928; JFM 54.0619.02)].
Mathematics Subject Classification ID: 54F45
zbMATH DE Number: 987922
zbMATH Keywords: dimension of product
zbMATH Keywords: stable intersection of compacta
zbMATH Keywords: JFM 54.0619.02
reviewed by: Dušan Repovš
MaRDI profile type: Publication
cites work: On compacta that intersect unstably in Euclidean space
full work available at URL: s0166-8641(96)00053-3
OpenAlex ID: W2000204055
description: scientific article; zbMATH DE number 987922