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The locally finite topology on $ 2\sp X$

Authors: G. A. Beer, C. J. Himmelberg, K. Prikry and F. S. Van Vleck
Journal: Proc. Amer. Math. Soc. 101 (1987), 168-172
MSC: Primary 54B20; Secondary 54A10
MathSciNet review: 897090
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Abstract: Let $ X$ be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace $ {2^X}$ of all closed, nonempty subsets of $ X$. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on $ X$. We also investigate when the locally finite topology coincides with the more usual topologies on $ {2^X}$ and when the locally finite topology is metrizable.

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Keywords: Hyperspaces, locally finite topology, Vietoris topology, Hausdorff metric topology, supremum topology, coincidences, UC space
Article copyright: © Copyright 1987 American Mathematical Society

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