Metric Compactifications and Coarse Structures

Mine, Kotaro ; Yamashita, Atsushi

Canadian journal of mathematics, Tome 67 (2015) no. 5, pp. 1091-1108

Résumé

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 $\widetilde{X}$ is determined only by the topology of the remainder $\widetilde{X}\,\backslash \,X$ .

Détail

DOI : 10.4153/CJM-2015-029-8

Mots-clés : 18B30, 51F99, 53C23, 54C20, coarse geometry, Higson corona, continuously controlled coarse structure, uniformcontinuity, boundary at infinity