In this text we will consider topologies on the closed subsets of a metric space and on the closed convex subsets of a normed linear space. We shall call such topological spaces hyperspaces. Many natural hyperspace topologies are not metrizable, and some are not even Hausdorff. Minimally, we will insist that a hyperspace topology extends the initial topology on the underlying metric space. In other words, if we restrict the topology to the singleton subsets, we want the induced subspace to agree with the initial topology on the underlying space. This property will be called admissibility.
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