On the dimension of \(\mathcal Z\)-sets (Q387232)

Recall that a closed subset \(A\) of an ANR \(Y\) is a \(\mathcal Z\)-set if there exists a homotopy \(H: Y\times [0,1] \to Y\) such that \(H_0 = \mathrm{id}_Y\) and \(H_t(Y) \subseteq Y-A\) for all \(t\in (0,1]\). \textit{M. Bestvina} and \textit{G. Mess} [J. Am. Math. Soc. 4, No. 3, 469--481 (1991; Zbl 0767.20014)] proved that, for a \(\mathcal Z\)-set \(A\) in a finite-dimensional ANR \(Y\), \(\dim A < \dim Y\). Their proof is based on the cohomological dimension, which coincides with the covering dimension for finite-dimensional spaces. In the paper under review, the authors give a short and elementary proof of this fact without using any knowledge of cohomological dimension or algebraic topology.