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Proceedings of the American Mathematical Society

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

 

 

Fine uniformity and the locally finite hyperspace topology


Authors: S. A. Naimpally and P. L. Sharma
Journal: Proc. Amer. Math. Soc. 103 (1988), 641-646
MSC: Primary 54B20; Secondary 54D30, 54E15
DOI: https://doi.org/10.1090/S0002-9939-1988-0943098-9
MathSciNet review: 943098
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Abstract: It is shown that a uniformizable space $ X$ is normal iff the locally finite topology $ {e^\tau }$ on the hyperspace $ {2^X}$ coincides with the topology transmitted by the fine uniformity of $ X$. We also prove that, for $ X$ normal, the topology $ {e^\tau }$ is first countable only if the set of limit points $ X'$ of $ X$ is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0943098-9
Keywords: Hyperspace, locally finite topology, fine uniformity, Hausdorff metric, pseudocompactness
Article copyright: © Copyright 1988 American Mathematical Society