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Keywords:
function spaces; Kuratowski convergence; hyperspaces
function spaces; Kuratowski convergence; hyperspaces
Summary:
Let $X$ be a locally connected, $b$-compact metric space and $E$ a closed subset of $X$. Let $\Bbb G$ be the space of all continuous real-valued functions defined on some closed subsets of $E$. We prove the equivalence of the ${\tau_{_{a\!w}}}$ and ${\tau^c_{_{\!K}}}$ topologies on $\Bbb G$, where $\tau_{_{a\!w}}$ is the so called {\sl Attouch-Wets\/} topology, defined in terms of uniform convergence of distance functionals, and ${\tau^c_{_{\!K}}}$ is the topology of Kuratowski convergence on compacta.
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