## Journal of the Mathematical Society of Japan

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

Katsuhiko MATSUZAKI

#### Abstract

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

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

Dates
First available in Project Euclid: 18 January 2017

https://projecteuclid.org/euclid.jmsj/1484730026

Digital Object Identifier
doi:10.2969/jmsj/06910263

Mathematical Reviews number (MathSciNet)
MR3597555

Zentralblatt MATH identifier
1371.54132

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1484730026

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