Abstract
It is shown that a uniformizable space X X is normal iff the locally finite topology e τ {e^\tau } on the hyperspace 2 X {2^X} coincides with the topology transmitted by the fine uniformity of X X . We also prove that, for X X normal, the topology e τ {e^\tau } is first countable only if the set of limit points X ′ X’ of X X is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given.
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