| Title:
|
Continuous selections on spaces of continuous functions (English) |
| Author:
|
Tamariz-Mascarúa, Ángel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
641-660 |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a space $Z$, we denote by $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ the hyperspaces of non-empty closed, compact, and subsets of cardinality $\leq 2$ of $Z$, respectively, with their Vietoris topology. For spaces $X$ and $E$, $C_p(X,E)$ is the space of continuous functions from $X$ to $E$ with its pointwise convergence topology. We analyze in this article when $\Cal{F}(Z)$, $\Cal{K}(Z)$ and $\Cal{F}_2(Z)$ have continuous selections for a space $Z$ of the form $C_p(X,E)$, where $X$ is zero-dimensional and $E$ is a strongly zero-dimensional metrizable space. We prove that $C_p(X,E)$ is weakly orderable if and only if $X$ is separable. Moreover, we obtain that the separability of $X$, the existence of a continuous selection for $\Cal{K}(C_p(X,E))$, the existence of a continuous selection for $\Cal{F}_2(C_p(X,E))$ and the weak orderability of $C_p(X,E)$ are equivalent when $X$ is $\Bbb{N}$-compact. Also, we decide in which cases $C_p(X,2)$ and $\beta C_p(X,2)$ are linearly orderable, and when $\beta C_p(X,2)$ is a dyadic space. (English) |
| Keyword:
|
continuous selections |
| Keyword:
|
Vietoris topology |
| Keyword:
|
linearly orderable space |
| Keyword:
|
weakly orderable space |
| Keyword:
|
space of continuous functions |
| Keyword:
|
dyadic spaces |
| MSC:
|
54B20 |
| MSC:
|
54C35 |
| MSC:
|
54C65 |
| MSC:
|
54F05 |
| idZBL:
|
Zbl 1150.54021 |
| idMR:
|
MR2337419 |
| . |
| Date available:
|
2009-05-05T17:00:18Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119625 |
| . |
| Reference:
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| . |
