Fine Uniformity and the Locally Finite Hyperspace Topology

S. A. Naimpally, P. L. Sharma
1988 Proceedings of the American Mathematical Society  
It is shown that a uniformizable space X is normal iff the locally finite topology eT on the hyperspace 2X coincides with the topology transmitted by the fine uniformity of X. We also prove that, for X normal, the topology eT 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.
doi:10.2307/2047193 fatcat:vp5o5vo5xnawzc3g454zkr75fe