Geodesic
Attouch-Wets convergence and Kuratowski convergence on compact sets
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 551-562.

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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.
Classification : 49J52, 54C35, 54C99
Keywords: function spaces; Kuratowski convergence; hyperspaces 
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     author = {Piccione, Paolo and Sampalmieri, Rosella},
     title = {Attouch-Wets convergence and {Kuratowski} convergence on compact sets},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {551--562},
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     zbl = {0844.54010},
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Piccione, Paolo; Sampalmieri, Rosella. Attouch-Wets convergence and Kuratowski convergence on compact sets. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 3, pp. 551-562. http://geodesic.mathdoc.fr/item/CMUC_1995__36_3_a16/