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Boundaries of coarse proximity spaces and boundaries of compactifications

Pawel Grzegrzolka, Jeremy Siegert

Abstract

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 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.

Boundaries of coarse proximity spaces and boundaries of compactifications

Abstract

In this paper, we introduce the boundary of a coarse proximity space This boundary is a subset of the boundary of a certain Smirnov compactification. We show that 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.

Paper Structure

This paper contains 9 sections, 36 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Proximities and the Smirnov Compactification
  3. Coarse Proximities
  4. Discrete Extensions and Boundaries of Coarse Proximity Spaces
  5. Coarse Proximities from Compactifications
  6. The Gromov Boundary
  7. The Visual Boundary
  8. The Higson Corona
  9. The Freudenthal Boundary

Key Result

Theorem \oldthetheorem

Let $(X,\delta)$ be a proximity space. Then $\delta$ induces a topology on $X$ defined by This topology is always completely regular, and is Hausdorff if and only if $\delta$ is separated.

Theorems & Definitions (111)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Example \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • ...and 101 more