  Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric
spaces with the $C_0$ coarse structures. We show that if $X$ and $Y$ are in
$\mathbf{TB}$ then $X$ and $Y$ are coarsely equivalent if and only if their
Higson coronas are homeomorphic. In fact, the Higson corona functor gives an
equivalence of categories $\mathbf{TB}\to\mathbf{K}$, where $\mathbf{K}$ is the
category of compact metrizable spaces. We use this fact to show that the
continuously controlled coarse structure on a locally compact space $X$ induced
by some metrizable compactification $\tilde{X}$ is determined only by the
topology of the remainder $\tilde{X}\setminus X$.
