Assouad-Nagata dimension and finite-to-one Lipschitz maps (Q2850639)

In 1926, Hurewicz characterized the covering dimension of a separable metric space by a closed map from a zero-dimensional space, as follows: for a separable metric space \(X\) and a nonnegative integer \(n\), \(\dim X\leq n\) if and only if there exists a closed map from a zero-dimensional space onto \(X\) such that \(f^{-1}(x)\) contains at most \(n+1\) points for each \(x\in X\). This result was generalized to classes of generalized metric spaces and Hurewicz-type theorems were obtained for other dimensional functions. In this paper, the authors prove a Hurewicz-type characterization of the Assouad-Nagata dimension.NEWLINENEWLINELet \((X,d)\) be a metric space and \(r>0\). A family \(\mathcal U\) of subsets of \(X\) is \(r\)-\textit{disjoint} if \(d(x,x')>r\) for any \(x\) and \(x'\) that belong to different elements of \(\mathcal U\). For every nonnegative integer \(n\), \(X\) is said to have \textit{Assouad-Nagata dimension at most \(n\)}, denoted \(\dim_{AN}X\leq n\), provided there exists a constant \(c>0\) such that for every \(r>0\), there exists a cover \({\mathcal U}=\bigcup\limits_{i=1}^{n+1}{\mathcal U}_i\) of \(X\) so that each \({\mathcal U}_i\) is \(r\)-disjoint and mesh\(\,(\mathcal U)\leq cr\). The authors obtain the following result: For a proper separable metric space \(X\) and a nonnegative integer, \(\dim_{AN}X\leq n\) if and only if there exist an ultrametric space \(Z\) and a Lipschitz map \(f\) from \(Z\) onto \(X\) such that \(f^{-1}(x)\) contains at most \(n+1\) points for each \(x\in X\) and they have the following property: there exists a constant \(c>0\) such that for each \(r>0\) and every subset \(B\) of \(X\) with diameter at most \(r\), there exists a subset \(A\) of \(Z\) with diameter at most \(cr\) and \(f(A)=B\).