Journal of the Mathematical Society of Japan

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

Katsuhiko MATSUZAKI

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730026

Digital Object Identifier
doi:10.2969/jmsj/06910263

Mathematical Reviews number (MathSciNet)
MR3597555

Zentralblatt MATH identifier
1371.54132

Subjects
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.)

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

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


Export citation

References

  • N. Bourbaki, Topologie générale, Diffusion C.C.L.S., Paris, 1971.
  • R. Canary, D. Epstein and P. Green, Notes on notes of Thurston, Fundamentals of Hyperbolic Manifolds, LMS Lecture Note Series, 328, Cambridge Univ. Press, 2006, pp.,1–115.
  • C. Charitos, I. Papadoperakis and A. Papadopoulos, On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface, Proc. Amer. Math. Soc., 142 (2014), 2179–2191.
  • P. de la Harpe, Spaces of closed subgroups of locally compact groups, arXiv:0807.2030v2.
  • J. Dugundji, Topology, Allyn and Bacon, 1966.
  • J. Kelley, General Topology, GTM, 27, Springer, 1975.
  • L. A. Steen and J. A. Seebach, Counterexamples in Topology, 2nd ed., Springer, 1978.
  • W. P. Thurston, The Geometry and Topology of Three-Manifolds, lecture note at Princeton Univ., 1980. http://www.msri.org/publications/books/gt3m/
  • S. Willard, General Topology, Addison-Wesley, 1970.
  • T. Yoshino, Topological blow-up and discontinuous groups, Representation Theory and Harmonic Analysis, Oberwolfach Report 52/2010, pp.,3076–3080.
  • T. Yoshino, On topological blow-up.