Abstract view
Metric Compactifications and Coarse Structures


Published:20150720
Printed: Oct 2015
Kotaro Mine,
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 1538914, Japan
Atsushi Yamashita,
Chiba Institute of Technology, 211, Shibazono, Narashinoshi, Chiba, 2750023, Japan
Abstract
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 $\tilde{X}\setminus
X$.