Geodesic
Kuratowski convergence on compacta and Hausdorff metric convergence on compacta
Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 2, pp. 309-318 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper completes and improves results of [10]. Let $(X,d_{_X})$, $(Y,d_{_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence $\tau_{_K}$ and the Kuratowski convergence on compacta $\tau_{_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology $\tau_{_{AW}}$ (generated by the box metric of $d_{_X}$ and $d_{_Y}$) and $\tau_{_K}^c$ convergence on $G$, which improves the main result of [10]. In the second part of paper we extend the definition of Hausdorff metric convergence on compacta for general metric spaces $X$ and $Y$ and we show that if $X$ is locally compact metric space, then also $\tau$-convergence and Hausdorff metric convergence on compacta coincide in $G$.
This paper completes and improves results of [10]. Let $(X,d_{_X})$, $(Y,d_{_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence $\tau_{_K}$ and the Kuratowski convergence on compacta $\tau_{_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology $\tau_{_{AW}}$ (generated by the box metric of $d_{_X}$ and $d_{_Y}$) and $\tau_{_K}^c$ convergence on $G$, which improves the main result of [10]. In the second part of paper we extend the definition of Hausdorff metric convergence on compacta for general metric spaces $X$ and $Y$ and we show that if $X$ is locally compact metric space, then also $\tau$-convergence and Hausdorff metric convergence on compacta coincide in $G$.
Classification : 54A20, 54B20, 54C35
Keywords: Kuratowski convergence; Attouch-Wets convergence; $\tau$-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta 
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     author = {Brandi, P. and Ceppitelli, R. and Hol\'a, \v{L}.},
     title = {Kuratowski convergence on compacta and {Hausdorff} metric convergence on compacta},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {309--318},
     year = {1999},
     volume = {40},
     number = {2},
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     zbl = {0976.54010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1999_40_2_a11/}
}
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Brandi, P.; Ceppitelli, R.; Holá, Ľ. Kuratowski convergence on compacta and Hausdorff metric convergence on compacta. Commentationes Mathematicae Universitatis Carolinae, Tome 40 (1999) no. 2, pp. 309-318. http://geodesic.mathdoc.fr/item/CMUC_1999_40_2_a11/