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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0402667-7
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.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0402667-7
Keywords: First countable, separable, hyperspaces, the finite topology, second countability
Article copyright: © Copyright 1976 American Mathematical Society