The Attouch-Wets and Hausdorff Metric Topologies
In the previous chapter we introduced the Wijsman topology on the closed subsets CL(X) of a metrizable space, which may be viewed as the topology that CL(X) inherits from (CX,R),equipped with the topology of pointwise convergence, under the identification A ↔ d(· ,A). By equicontinuity of distance functionals, no stronger subspace topology results if the compact-open topology replaces the topology of pointwise convergence. In this chapter, we consider stronger (metrizable) topologies on C(X,R) and the topologies they induce on CL(X) : the topology of uniform convergence and the topology of uniform convergence on bounded subsets of X. The resulting hyperspace topologies on CL(X) are called the Hausdorff metric topology and the Attouch-Wets topology.
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