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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