The Chabauty and the Thurston topologies on the hyperspace of closed subsets

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Abstract

For a regularly locally compact topological space X of T0 separation axiom but not necessarily Hausdorff, we consider a map σ from X to the hyperspace C(X) of all closed subsets of X by taking the closure of each point of X. By providing the Thurston topology for C(X), we see that σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification X̂ of X. In this paper, we investigate properties of X̂ and C(X̂) equipped with different topologies. In particular, we consider a condition under which a self-homeomorphism of a closed subspace of C(X) with respect to the Chabauty topology is a self-homeomorphism in the Thurston topology.

Original languageEnglish
Pages (from-to)263-292
Number of pages30
JournalJournal of the Mathematical Society of Japan
Volume69
Issue number1
DOIs
Publication statusPublished - 2017

Keywords

  • Chabauty topology
  • Compactification
  • Filter
  • Geodesic lamination
  • Hausdorff space
  • Hyperspace
  • Locally compact
  • Net
  • Thurston topology

ASJC Scopus subject areas

  • Mathematics(all)

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