| Title:
|
Disconnectedness properties of hyperspaces (English) |
| Author:
|
Hernández-Gutiérrez, Rodrigo |
| Author:
|
Tamariz-Mascarúa, Angel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2011 |
| Pages:
|
569-591 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ be a Hausdorff space and let $\mathcal H$ be one of the hyperspaces $CL(X)$, $\mathcal K(X)$, $\mathcal F(X)$ or $\mathcal F_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal H$: extremal disconnectedness, being a $F'$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that (a) both $F$ and $X-F$ are totally disconnected, (b) the quotient $X/F$ is hereditarily disconnected, then $\mathcal K(X)$ is hereditarily disconnected. We also show an example proving that this result cannot be reversed. (English) |
| Keyword:
|
hyperspaces |
| Keyword:
|
Vietoris topology |
| Keyword:
|
$F'$-space |
| Keyword:
|
$P$-space |
| Keyword:
|
hereditarily disconnected |
| MSC:
|
54B20 |
| MSC:
|
54G05 |
| MSC:
|
54G10 |
| MSC:
|
54G12 |
| MSC:
|
54G20 |
| idZBL:
|
Zbl 1249.54024 |
| idMR:
|
MR2864000 |
| . |
| Date available:
|
2011-12-16T13:52:33Z |
| Last updated:
|
2015-02-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141742 |
| . |
| Reference:
|
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