Metric compactifications and coarse structures

Details Metric compactifications and coarse structures Metric compactifications and coarse structures

2011-06-08

arxiv.org/abs/1106.1672v3

Mathematics

General Topology

16 pages. v3: Corrected typos. v2: Title changed from "$C_0$ coarse structures and Smirnov compactifications". Considerable im

Scientific paper

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 $\tilde{X}\setminus X$.

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