Journal of the Mathematical Society of Japan

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


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For a regularly locally compact topological space $X$ of $\rm T_0$ separation axiom but not necessarily Hausdorff, we consider a map $\sigma$ 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 $\sigma$ is a topological embedding, and by taking the closure of $\sigma(X)$ with respect to the Chabauty topology, we have the Hausdorff compactification $\widehat X$ of $X$. In this paper, we investigate properties of $\widehat X$ and $C(\widehat X)$ 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.

Article information

J. Math. Soc. Japan, Volume 69, Number 1 (2017), 263-292.

First available in Project Euclid: 18 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
Secondary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 54D10: Lower separation axioms (T0-T3, etc.) 54D35: Extensions of spaces (compactifications, supercompactifications, completions, etc.)

hyperspace Chabauty topology Thurston topology filter net Hausdorff space compactification locally compact geodesic lamination


MATSUZAKI, Katsuhiko. The Chabauty and the Thurston topologies on the hyperspace of closed subsets. J. Math. Soc. Japan 69 (2017), no. 1, 263--292. doi:10.2969/jmsj/06910263.

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