Metric compactifications and coarse structures

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 $\tilde{X}\setminus X$.

16 pages. v3: Corrected typos. v2: Title changed from "$C_0$ coarse structures and Smirnov compactifications". Considerable improvements in the proofs of the main theorem (Theorem 4.5) and an important lemma (Lemma 3.7). New results in the final section (Theorem 4.12 among others)

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• A $C_0$ coarse structure for families of pseudometrics and the Higson-Roe functor
• Coarse metric and uniform metric