Fine Uniformity and the Locally Finite Hyperspace Topology

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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.