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Uniform continuity on bounded sets and the Attouch-Wets topology


Authors: Gerald Beer and Anna Di Concilio
Journal: Proc. Amer. Math. Soc. 112 (1991), 235-243
MSC: Primary 54B20
DOI: https://doi.org/10.1090/S0002-9939-1991-1033956-1
MathSciNet review: 1033956
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Abstract: Let $ {\text{CL(}}X{\text{)}}$ be the nonempty closed subsets of a metrizable space $ X$. If $ d$ is a compatible metric, the metrizable Attouch-Wets topology $ {\tau _{aw}}(d)$ on $ {\text{CL(}}X{\text{)}}$ is the topology of uniform convergence of distance functionals associated with elements of $ {\text{CL(}}X{\text{)}}$ on bounded subsets of $ X$. The main result of this paper shows that two compatible metrics $ d$ and $ \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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1033956-1
Keywords: Attouch-Wets topology, uniform continuity on bounded sets, uniform convergence of distance functionals on bounded sets, set convergence, hyperspace
Article copyright: © Copyright 1991 American Mathematical Society

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