Kuratowski convergence on compacta and Hausdorff metric convergence on compacta

Brandi, Primo, Ceppitelli, Rita, and Holá, Ľubica. "Kuratowski convergence on compacta and Hausdorff metric convergence on compacta." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 309-318. <http://eudml.org/doc/248381>.

@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 -