Kuratowski convergence on compacta and Hausdorff metric convergence on compacta

@article{Brandi1999,
abstract = {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 $G$.},
author = {Brandi, Primo, Ceppitelli, Rita, Holá, Ľubica},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Kuratowski convergence; Attouch-Wets convergence; $\tau $-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta; -convergence; Kuratowski convergence on compacta; Hausdorff metric convergence on compacta},
language = {eng},
number = {2},
pages = {309-318},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Kuratowski convergence on compacta and Hausdorff metric convergence on compacta},
url = {http://eudml.org/doc/248381},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Brandi, Primo
AU - Ceppitelli, Rita
AU - Holá, Ľubica
TI - Kuratowski convergence on compacta and Hausdorff metric convergence on compacta
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 309
EP - 318
AB - 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 $G$.
LA - eng
KW - Kuratowski convergence; Attouch-Wets convergence; $\tau $-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta; -convergence; Kuratowski convergence on compacta; Hausdorff metric convergence on compacta
UR - http://eudml.org/doc/248381
ER -