Uniform continuity on bounded sets and the Attouch-Wets topology

Beer, Gerald; Di Concilio, Anna

doi:10.1090/S0002-9939-1991-1033956-1

Uniform continuity on bounded sets and the Attouch-Wets topology
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by Gerald Beer and Anna Di Concilio PDF

Proc. Amer. Math. Soc. 112 (1991), 235-243 Request permission

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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