Boundaries of coarse proximity spaces and boundaries of compactifications
In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ 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 compact Hausdorff space can be realized as the boundary of a coarse proximity space. We then show that many boundaries of well-known compactifications arise naturally as boundaries of coarse proximity spaces. In particular, we give 4 natural coarse proximity structures whose boundaries are the Higson, Freudenthal, Gromov, and Visual boundary, respectively.