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Metric Compactifications and Coarse Structures

Published:2015-07-20
Printed: Oct 2015
• Kotaro Mine,
Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo 153-8914, Japan
• Atsushi Yamashita,
Chiba Institute of Technology, 2-1-1, Shibazono, Narashino-shi, Chiba, 275-0023, Japan
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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 $\tilde{X}\setminus X$.
 Keywords: coarse geometry, Higson corona, continuously controlled coarse structure, uniform continuity, boundary at infinity
 MSC Classifications: 18B30 - Categories of topological spaces and continuous mappings [See also 54-XX] 51F99 - None of the above, but in this section 53C23 - Global geometric and topological methods (a la Gromov); differential geometric analysis on metric spaces 54C20 - Extension of maps

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