A short elementary proof of the Dranishnikov-West theorem on stable intersection of compacta in Euclidean spaces
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)].
scientific article; zbMATH DE number 987922