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A note on completeness of hyperspaces. (English) Zbl 0766.54008
General topology and applications, Proc. 5th Northeast Conf., New York/NY (USA) 1989, Lect. Notes Pure Appl. Math. 134, 19-24 (1991).
Summary: [For the entire collection see Zbl 0744.00027.]
The hyperspace of a uniform space, as defined by Bourbaki, satisfies certain properties if and only if the base space does. Examples of such properties would be metrizability and compactness. Completeness, however, is not one of these properties. Isbell gave necessary and sufficient conditions for a uniform space to have a complete hyperspace, and one of these conditions made use of functions, defined on partially ordered sets, and taking values in the space. Here we give a characterization of spaces with complete hyperspaces which is stated solely in terms of nets into the space. Our condition is closely related to the property of cofinal completeness, due to N. R. Howes [Pac. J. Math. 38, 431-440 (1971; Zbl 0202.540)].

54B20 Hyperspaces in general topology
54E15 Uniform structures and generalizations