A new construction of universal spaces for asymptotic dimension

For each n  , we construct a separable metric space UnUn that is universal in the coarse category of separable metric spaces with asymptotic dimension (asdim) at most n and universal in the uniform category of separable metric spaces with uniform dimension (udim) at most n  . Thus, UnUn serves as a universal space for dimension n in both the large-scale and infinitesimal topology. More precisely, we prove:asdimUn=udimUn=nasdimUn=udimUn=n and such that for each separable metric space X,a)if asdimX⩽n, then X   is coarsely equivalent to a subset of UnUn;b)if udimX⩽n, then X   is uniformly homeomorphic to a subset of UnUn.