@article{ff197d5860ab4ff294908ff9916aa50d,

title = "The Chabauty and the Thurston topologies on the hyperspace of closed subsets",

abstract = "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 {\^X} of X. In this paper, we investigate properties of {\^X} and C({\^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.",

keywords = "Chabauty topology, Compactification, Filter, Geodesic lamination, Hausdorff space, Hyperspace, Locally compact, Net, Thurston topology",

author = "Katsuhiko Matsuzaki",

note = "Funding Information: This work was supported by JSPS KAKENHI (No. 24654035). The author would like to thank the referee for reading the entire manuscript very carefully and correctly. Because of this contribution, the paper made much progress on its representation and several mistakes in the previous versions were corrected. Among them, the following important changes are due to his/her point out: (1) We formulate Proposition 2.7 based on the referee's comment. (2) In Lemma 9.2 and other results using this lemma, we put the condition that not only the bijection {\~f} : {\~X}→Ŷ but also {\~f}-1 preserve the inclusion relation and the point structure; (3) We correct the proof of Theorem 9.3 in which we tried to show that t : X → C({\^X})T was a topological embedding. (4) In the proof of Theorem 7.4, we show that {\^f}-1(O2(K)∩Ŷ) is open by the referee's argument. (5) We notice that the condition that {\~f} : {\~X} → Ỹ is surjective is necessary in Proposition 8.5, and add this assumption also to Theorem 8.6. Publisher Copyright: {\textcopyright}2017 The Mathematical Society of Japan.",

year = "2017",

doi = "10.2969/jmsj/06910263",

language = "English",

volume = "69",

pages = "263--292",

journal = "Journal of the Mathematical Society of Japan",

issn = "0025-5645",

publisher = "Mathematical Society of Japan - Kobe University",

number = "1",

}