A new construction of universal spaces for asymptotic dimension

For each n, we construct a separable metric space U n 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, U n serves as a universal space for dimension n in both the large-scale and infinitesimal topology. More precisely, we prove: asdim U n = udim U n = n and such that for each separable metric space X,a) im X ⩽ n , then X is coarsely equivalent to a subset of U n ; m X ⩽ n , then X is uniformly homeomorphic to a subset of U n .