Fine uniformity and the locally finite hyperspace topology

Naimpally, S. A.; Sharma, P. L.

doi:10.1090/S0002-9939-1988-0943098-9

Fine uniformity and the locally finite hyperspace topology
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by S. A. Naimpally and P. L. Sharma PDF

Proc. Amer. Math. Soc. 103 (1988), 641-646 Request permission

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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