### Abstract

For a regularly locally compact topological space X of T_{0} separation axiom but not necessarily Hausdorff, we consider a map σ 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 σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification X̂ of X. In this paper, we investigate properties of X̂ and C(X̂) 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.

Original language | English |
---|---|

Pages (from-to) | 263-292 |

Number of pages | 30 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 69 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

### Fingerprint

### Keywords

- Chabauty topology
- Compactification
- Filter
- Geodesic lamination
- Hausdorff space
- Hyperspace
- Locally compact
- Net
- Thurston topology

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The Chabauty and the Thurston topologies on the hyperspace of closed subsets.** / Matsuzaki, Katsuhiko.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The Chabauty and the Thurston topologies on the hyperspace of closed subsets

AU - Matsuzaki, Katsuhiko

PY - 2017

Y1 - 2017

N2 - For a regularly locally compact topological space X of T0 separation axiom but not necessarily Hausdorff, we consider a map σ 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 σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification X̂ of X. In this paper, we investigate properties of X̂ and C(X̂) 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.

AB - For a regularly locally compact topological space X of T0 separation axiom but not necessarily Hausdorff, we consider a map σ 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 σ is a topological embedding, and by taking the closure of σ(X) with respect to the Chabauty topology, we have the Hausdorff compactification X̂ of X. In this paper, we investigate properties of X̂ and C(X̂) 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.

KW - Chabauty topology

KW - Compactification

KW - Filter

KW - Geodesic lamination

KW - Hausdorff space

KW - Hyperspace

KW - Locally compact

KW - Net

KW - Thurston topology

UR - http://www.scopus.com/inward/record.url?scp=85013658188&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85013658188&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06910263

DO - 10.2969/jmsj/06910263

M3 - Article

AN - SCOPUS:85013658188

VL - 69

SP - 263

EP - 292

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -