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Confinal completeness of the Hausdorff metric topology. (English summary) Fund. Math. 208 (2010), no. 1, 75–85.1730-6329 Let (X, d) be a metric space, and let C0(X) denote the family of all nonempty closed subsets of (X, d). For any A, B ∈ C0(X), the Hausdorff distance between A and B is defined by Hd(A, B) = max{sup{d(a, B) : a ∈ A}, sup{d(b, A) : b ∈ B}}. The Hausdorff distance so defined is an extended real-valued metric on C0(X) which is finitevalued when restricted to the nonempty closed and bounded sets. We shall call (C0(X), Hd) the hyperspace of (X, d). It is well known that the properties of completeness, total boundedness, and compactness of (X, d) carry over to the hyperspace (C0(X), Hd). In the paper under review, the authors provide some necessary and sufficient conditions for (C0(X), Hd) to be cofinally complete. To state the main result of the paper, we need to recall some definitions. A net {xλ: λ ∈ Λ} in a Hausdorff uniform space (X, U) is called cofinally Cauchy, if for each entourage U ∈ U there exists a cofinal subset Λ0 ⊆ Λ such that (xλ, xµ) ∈ U whenever λ, µ ∈ Λ0, and (X, U) is called cofinally complete if each cofinally Cauchy net clusters. Cofinal completeness was used to characterize paracompact spaces by H. H. Corson [Amer. J. Math. 8

Year: 2013
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