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Keywords:
Kuratowski convergence; Attouch-Wets convergence; $\tau$-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta
Kuratowski convergence; Attouch-Wets convergence; $\tau$-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta
Summary:
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 $G$.
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