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By Reviewed Jiling Cao


Given a topological space X, let CL∅(X) denote the collection of all closed subsets of X, and let CL(X) = CL∅(X) � {∅}. A well-known theorem of V. I. Malyhin [Dokl. Akad. Nauk SSSR 203 (1972), 1001–1003; MR0300241 (45 #9287)] establishes that for the hyperspace CL(X) of a normal space X endowed with the Vietoris topology, being first countable, Fréchet, sequential or countably tight are equivalent. But, when CL(X) is endowed with either the Fell topology, or with its upper or lower parts, the conclusion of Malyhin’s theorem is no longer true. This motivates the authors of the paper under review to investigate the character, tightness, Fréchet property and sequentiality of CL∅(X) or CL(X), with the co-compact, the lower Vietoris and the Fell topologies, respectively. The authors begin their investigation by studying some cardinal functions on the base space X. In the subsequent sections, those cardinal functions are used to calculate or estimate the character and tightness of the hyperspace CL∅(X) or CL(X). Furthermore, radiality and pseudoradiality of CL∅(X) or CL(X) are considered as well. The main tool employed in this paper is the “decomposition ” technique in hyperspace theory. Some results in [J.-C. Hou, Topology Appl. 84 (1998), no. 1-3, 199–206; MR1611218 (99a:54004)] are improved. In addition, the authors give some examples, which may be of certain independent interest, and also pose several interesting questions

Year: 2013
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