Title: A new construction of universal spaces for asymptotic dimension,
Abstract: 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 ($\mathop{asdim}$) at most $n$ and universal in the uniform category
of separable metric spaces with uniform dimension ($\mathop{udim}$) 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:
\[
\mathop{asdim} \mathbb{U}_n = \mathop{udim} \mathbb{U}_n = n
\] and such that for each separable metric space $X$,
a) if $\mathop{asdim} X \leq n$, then $X$ is coarsely equivalent to a subset
of $\mathbb{U}_n$;
b) if $\mathop{udim} X \leq n$, then $X$ is uniformly homeomorphic to a
subset of $\mathbb{U}_n$.