A new construction of universal spaces for asymptotic dimension (Q1928428)

Since this paper involves the uniform covering dimension (udim) and the asymptotic dimension (asdim) of a metric space, we list the definitions of these concepts. We assume that \(X\) is a metric space and \(n\) is an integer. The uniform covering dimension udim\(X\) of \(X\) is less than or equal \(n\) if for each \(r>0\), there exists an open cover of \(X\) with mesh smaller than \(r\), having a positive Lebesgue number, and being of multiplicity at most \(n+1\). The asymptotic dimension asdim\(X\) of \(X\) is less than or equal \(n\) if for each \(r<\infty\), there exists an open cover of \(X\) with Lebesgue number greater than \(r\), having finite mesh, and being of multiplicity at most \(n+1\). The results herein are for separable metric spaces. In previous work [\textit{A. Dranishnikov} and \textit{M. Zarichnyi}, Topology Appl. 140, No. 2--3, 203--225 (2004; Zbl 1063.54027)], there was given a construction of a space that is universal in the coarse category of separable metric spaces with asymptotic dimension at most \(n\). The work in the current paper aims ``to give a more transparent construction that highlights the micro-macro analogy between small and large scales.'' The authors prove that for each \(n\), there is a separable metric space \(\mathbb{U}_n\) that is universal in the coarse category of separable metric spaces with asdim at most \(n\) and simultaneously universal in the uniform category of separable metric spaces with udim at most \(n\). This means that asdim\(\mathbb{U}_n=n=\)udim\(\mathbb{U}_n\), and for each separable metric space \(X\), 1) if asdim\(X\leq n\), then \(X\) is coarsely equivalent to a subspace of \(\mathbb{U}_n\); 2) if udim\(X\leq n\), then \(X\) is uniformly homeomorphic to a subspace of \(\mathbb{U}_n\).