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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



First countable hyperspaces

Author: R. E. Smithson
Journal: Proc. Amer. Math. Soc. 56 (1976), 325-328
MSC: Primary 54B20
MathSciNet review: 0402667
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Abstract: An example is given which shows that the space of compact subsets $ \mathcal{K}(X)$ of a first countable space $ X$ need not be first countable in the finite topology. Further, it is shown that if $ \mathcal{K}(X)$ is first countable then each compact subset of $ X$ is separable. Finally a characterization of $ \mathcal{K}(X)$ first countable in terms of a weak second countability condition is derived.

References [Enhancements On Off] (What's this?)

  • [1] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
  • [2] L. A. Steen and J. A. Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, New York, 1970. MR 42 #1040. MR 0266131 (42:1040)

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Keywords: First countable, separable, hyperspaces, the finite topology, second countability
Article copyright: © Copyright 1976 American Mathematical Society

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