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Coarse Proximity and Proximity at Infinity

Pawel Grzegrzolka, Jeremy Siegert

Abstract

We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space $X$ is a coarse invariant of $X$.

Coarse Proximity and Proximity at Infinity

Abstract

We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space is a coarse invariant of .

Paper Structure

This paper contains 9 sections, 47 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric Coarse Proximity
  4. Coarse Proximity Spaces
  5. Coarse Neighborhoods
  6. Equivalence Relation Induced by Coarse Proximity
  7. Coarse proximity maps
  8. Proximity at Infinity
  9. Questions

Key Result

Proposition \oldthetheorem

Given a function $f:X\rightarrow (Y,\delta_{2})$, the coarsest proximity $\delta_{0}$ on $X$ for which $f$ is a proximity map is defined by

Figures (1)

  • Figure 1: Construction of $E$

Theorems & Definitions (158)

  • Definition \oldthetheorem
  • Example \oldthetheorem
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  • Remark \oldthetheorem
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  • ...and 148 more