An embedding theorem for the Fell topology
Let X be a metric space, and let CL(X) be the set of all nonempty closed subsets of X. It is known that, when the metric d on X is bounded, \(E\mapsto d(\cdot,E)\) defines an embedding of CL(X) with the Hausdorff metric topology into the space of bounded continuous real-valued functions on X with the topology of uniform convergence. (In fact, it is an isometry with respect to the Hausdorff metric induced by d and the uniform metric.) This paper determines when the same formula gives an embedding of CL(X) with the weaker Fell topology [defined in the preceding review] into the space of continuous real-valued functions on X with the topology of uniform convergence on compacta. What is needed is a metric with respect to which X has nice closed balls (i.e., each noncompact closed ball is all of X), and such a metric is admissible iff X is locally compact. This result provides an alternate route to known facts about the Fell topology.
topology of uniform convergence on compacta
scientific article; zbMATH DE number 4078217