Boundaries of coarse proximity spaces and boundaries of compactifications

Abstract

In this paper, we introduce the boundary UX of a coarse proximity space (X, B, b). This boundary is a subset of the boundary of a certain Smirnov compactification. We show that UX 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. Contents1. Introduction 1 2. Coarse Proximities 3 3. Clusters in Proximity Spaces and the Smirnov Compactification 5 4. Discrete Extensions and Boundaries of Coarse Proximity Spaces 8 5. Compatibility of Proximities and Coarse Proximities 14 6. Compact Hausdorff Spaces as Boundaries of Coarse Proximity Spaces 17 7. The Higson Boundary 22 8. The Freudenthal Boundary 23 9. The Gromov Boundary 27 10. The Visual Boundary 29 References 31

Coarse Homotopy on metric Spaces and their Corona

“…This implies there is a subset S ⊆ A with f (S) g(S). Now by [6,Proposition 4.7] there is a coarse ultrafilter F on X with S ∈ F. Then f (S) ∈ f * F and g(S) ∈ g * F . Since f (S) g(S) this implies f * F = g * F .…”

“…The corona ν ′ (X) of a metric space X has been introduced in [1] and studied in [2], [3], [4], [5], [6], [7].…”

Inductive dimensions of coarse proximity spaces

“…We call the subset X \ X the Smirnov boundary of X. Now let us recall basic definitions and theorems surrounding coarse proximity spaces, as found in [9]. Definition 2.4.…”

“…As it was shown in [9], given a coarse proximity space (X, B, b), the boundary space associated to X is defined using the following proximity associated to a coarse proximity:…”

“…Coarse proximity structures lie between metric spaces and coarse spaces (as defined by Roe in [15]) in a way similar to how proximity spaces relate to metric spaces and uniform spaces (see [11]). In [9], the authors construct a functor from the category of coarse proximity spaces to the category of compact Hausdorff spaces that assigns to each coarse proximity space a certain "boundary space." This functor provides a common language for speaking of boundary spaces such as the Higson corona, the Gromov boundary, and other well-known boundary spaces.…”