Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 69, Number 1 (2017), 263-292.
The Chabauty and the Thurston topologies on the hyperspace of closed subsets
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.
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 263-292.
First available in Project Euclid: 18 January 2017
Permanent link to this document
Digital Object Identifier
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.)
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