A new construction of universal spaces for asymptotic dimension

For each $n$, we construct a separable metric space ${\mathbb{U}}_n$ that is universal in the coarse category of separable metric spaces with asymptotic dimension ($$) at most $n$ and universal in the uniform category of separable metric spaces with uniform dimension ($$) at most $n$. Thus, ${\mathbb{U}}_n$ serves as a universal space for dimension $n$ in both the large-scale and infinitesimal topology. More precisely, we prove: $${\mathbb{U}}_n={\mathbb{U}}_n=n$$ and such that for each separable metric space $X$, a) if $X\le n$, then $X$ is coarsely equivalent to a subset of ${\mathbb{U}}_n$; b) if $X\le n$, then $X$ is uniformly homeomorphic to a subset of ${\mathbb{U}}_n$.