Boundaries of coarse proximity spaces and boundaries of compactifications

Abstract

In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},\mathbf{b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and Hausdorff and that every compactification of a locally compact Hausdorff space induces a coarse proximity structure whose corresponding boundary is the boundary of the compactification. We then show that many boundaries of well-known compactifications arise as boundaries of coarse proximity spaces. In particular, we give four coarse proximity structures whose boundaries are the Gromov, visual, Higson, and Freudenthal boundaries.

arXiv comment27 pages

References in corpus (4)

• Asymptotic topology
• Coarse Proximity and Proximity at Infinity
• Metric compactifications and coarse structures
• Normality conditions of structures in coarse geometry and an alternative description of coarse proximities

Cited by in corpus (2)

• Coarse Homotopy on metric Spaces and their Corona
• Some counterexamples in abstract coarse geometry