Abstract
Let CL( X ) {\text {CL(}}X{\text {)}} be the nonempty closed subsets of a metrizable space X X . If d d is a compatible metric, the metrizable Attouch-Wets topology τ a w ( d ) {\tau _{aw}}(d) on CL( X ) {\text {CL(}}X{\text {)}} is the topology of uniform convergence of distance functionals associated with elements of CL( X ) {\text {CL(}}X{\text {)}} on bounded subsets of X X . The main result of this paper shows that two compatible metrics d d and ρ \rho determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continuous on bounded sets.
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