Coarse proximity and proximity at infinity

Pawel Grzegrzolka, Jeremy Siegert

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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.

Original languageEnglish
Pages (from-to)18-46
Number of pages29
JournalTopology and its Applications
Volume251
DOIs
StatePublished - 1 Jan 2019
Externally publishedYes

Keywords

  • Coarse geometry
  • Coarse proximity
  • Coarse topology
  • Hyperspace
  • Metric geometry
  • Proximity
  • Proximity at infinity

ASJC Scopus subject areas

  • Geometry and Topology

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