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The Chabauty and the Thurston topologies on the hyperspace of closed subsets. (English) Zbl 1371.54132

In this paper, the author studies the Chabauty topology and the Thurston topology on the set of closed subsets of (not necessarily Hausdorff) topological spaces. In particular, the author describes the topological blow-up due to Yoshino in terms of these topologies and gives an application to the space of geodesic laminations on a complete hyperbolic surface.
Let \(X\) be a topological space and \(C(X)\) the set of all closed subsets of \(X\). The Chabauty topology (or Fell topology) on \(C(X)\) is induced by the sub-basis consisting of all sets of the form \(\{ \alpha \in C(X) : \alpha \cap U \neq \emptyset\}\), where \(U\) is an open subset of \(X\), and all sets of the form \(\{ \alpha \in C(X) : \alpha \cap K = \emptyset\}\), where \(K\) is a compact subset of \(X\). The Thurston topology (or lower semi-finite topology, lower Vietoris topology) on \(C(X)\) has the sub-basis consisting of all sets of the form \(\{ \alpha \in C(X) : \alpha \cap U \neq \emptyset\}\), where \(U\) is an open subset of \(X\).
One of the theorems of this paper is the following: Let \(X\) be a \(T_0\)-space in which every point has a neighborhood basis consisting of compact subsets, and let \(\sigma : X \to C(X)\) be the mapping defined by \(\sigma(x)\) being the closure of \(\{x\}\) for each \(x \in X\). Then \(\sigma\) is a topological embedding with respect to the Thurston topology on \(C(X)\) and the closure \(\overline{\sigma(X)}^{\text{CH}}\) of the image \(\sigma(X)\) with respect to the Chabauty topology is a compact Hausdorff space (as a subspace of the Chabauty topology). Moreover, the closure \(\overline{\sigma(X)}^{\text{CH}}\) coincides with the topological blow-up of \(X\), which was introduced by Yoshino in order to turn a non-Hausdorff space into a Hausdorff one. A related notion (called the recovering map) is also studied.
Let \(X\) be a complete hyperbolic surface (not necessarily of finite area). A geodesic lamination of \(X\) is a closed subset of \(X\) that is a disjoint union of simple closed or infinite geodesics. Let \(\mathcal{GL}(X)\) be the set of all geodesic laminations of \(X\), and let \(\mathcal{GL}(X)_{\text{CH}}\) and \(\mathcal{GL}(X)_{\text{T}}\) be the subspaces of \(C(X)\) with the Chabauty topology and the Thurston topology, respectively. A map \(f : \mathcal{GL}(X) \to \mathcal{GL}(X)\) is said to preserve the inclusion relation if for any \(\lambda, \lambda^* \in \mathcal{GL}(X)\), \(\lambda \subset \lambda^*\) if and only if \(f(\lambda) \subset f(\lambda^*)\). The author also proves that a bijection \(f : \mathcal{GL}(X)_{\text{T}} \to \mathcal{GL}(X)_{\text{T}}\) is a homeomorphism if and only if \(f : \mathcal{GL}(X)_{\text{CH}} \to \mathcal{GL}(X)_{\text{CH}}\) is a homeomorphism such that \(f\) and \(f^{-1}\) preserve the inclusion relation.

MSC:

54B20 Hyperspaces in general topology
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D10 Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
57M50 General geometric structures on low-dimensional manifolds
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