| Title:
|
Kuratowski convergence on compacta and Hausdorff metric convergence on compacta (English) |
| Author:
|
Brandi, P. |
| Author:
|
Ceppitelli, R. |
| Author:
|
Holá, Ľ. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
40 |
| Issue:
|
2 |
| Year:
|
1999 |
| Pages:
|
309-318 |
| . |
| Category:
|
math |
| . |
| 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 $G$. (English) |
| Keyword:
|
Kuratowski convergence |
| Keyword:
|
Attouch-Wets convergence |
| Keyword:
|
$\tau$-convergence |
| Keyword:
|
Kuratowski convergence on compacta and Hausdorff metric convergence on compacta |
| MSC:
|
54A20 |
| MSC:
|
54B20 |
| MSC:
|
54C35 |
| idZBL:
|
Zbl 0976.54010 |
| idMR:
|
MR1732651 |
| . |
| Date available:
|
2009-01-08T18:52:20Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119086 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[4] Brandi P., Ceppitelli R.: A new graph topology. Connections with compact open topology.Appl. Analysis 53 (1994), 185-196. MR 1379407 |
| Reference:
|
[5] Brandi P., Ceppitelli R.: A new graph topology intended for functional differential equations.Atti Sem. Mat. Univ. Modena 54 (1996), 43-52. Zbl 0890.54010, MR 1405228 |
| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] Holá L'.: The Attouch-Wets topology and a characterization of normable linear spaces.Bull. Austral. Math. Soc. 44 (1991), 11-18. MR 1120389 |
| Reference:
|
[10] Piccione P., Sampalmieri R.: Attouch-Wets convergence and Kuratowski convergence on compact sets.Comment. Math. Univ. Carolinae 36 (1995), 551-562. Zbl 0844.54010, MR 1364496 |
| Reference:
|
[11] Sampalmieri R.: Kuratowski convergence on compact sets.Atti Sem. Mat. Fis. Univ. Modena 39 (1992), 381-390. Zbl 0770.54016, MR 1200296 |
| . |
