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January, 2017 The Chabauty and the Thurston topologies on the hyperspace of closed subsets
J. Math. Soc. Japan 69(1): 263-292 (January, 2017). DOI: 10.2969/jmsj/06910263


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.


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Katsuhiko MATSUZAKI. "The Chabauty and the Thurston topologies on the hyperspace of closed subsets." J. Math. Soc. Japan 69 (1) 263 - 292, January, 2017.


Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 1371.54132
MathSciNet: MR3597555
Digital Object Identifier: 10.2969/jmsj/06910263

Primary: 54A10
Secondary: 54A20 , 54D10 , 54D35

Keywords: Chabauty topology , compactification , filter‎ , geodesic lamination , Hausdorff space , hyperspace , Locally Compact , net , Thurston topology

Rights: Copyright © 2017 Mathematical Society of Japan


Vol.69 • No. 1 • January, 2017
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