| Title:
|
Weak selections and weak orderability of function spaces (English) |
| Author:
|
Gutev, Valentin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
1 |
| Year:
|
2010 |
| Pages:
|
273-281 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is proved that for a zero-dimensional space $X$, the function space $C_p(X,2)$ has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if $X$ is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space $E$, the function space $C_p(X,E)$ is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel. (English) |
| Keyword:
|
Vietoris hyperspace |
| Keyword:
|
continuous selection |
| Keyword:
|
function space |
| Keyword:
|
weakly orderable space |
| MSC:
|
54B20 |
| MSC:
|
54C35 |
| MSC:
|
54C65 |
| MSC:
|
54F05 |
| idZBL:
|
Zbl 1224.54026 |
| idMR:
|
MR2595088 |
| . |
| Date available:
|
2010-07-20T16:34:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140567 |
| . |
| Reference:
|
[1] García-Ferreira, S., Gutev, V., Nogura, T.: Extensions of 2-point selections.New Zealand J. Math. 38 (2008), 1-8. MR 2491681 |
| Reference:
|
[2] Gutev, V.: Weak orderability of second countable spaces.Fund. Math. 196 (2007), 275-287. Zbl 1129.54016, MR 2353859, 10.4064/fm196-3-4 |
| Reference:
|
[3] Gutev, V., Nogura, T.: Selections and order-like relations.Appl. Gen. Topol. 2 (2001), 205-218. Zbl 0993.54019, MR 1890037, 10.4995/agt.2001.2150 |
| Reference:
|
[4] Gutev, V., Nogura, T.: Vietoris continuous selections and disconnectedness-like properties.Proc. Amer. Math. Soc. 129 (2001), 2809-2815. Zbl 0973.54021, MR 1838807, 10.1090/S0002-9939-01-05883-X |
| Reference:
|
[5] Gutev, V., Nogura, T.: Selection problems for hyperspaces.Open Problems in Topology 2 (Elliott Pearl, ed.), Elsevier BV, Amsterdam (2007), 161-170. MR 2367385 |
| Reference:
|
[6] Hrušák, M., Martínez-Ruiz, I.: Selections and weak orderability.Fund. Math. 203 (2009), 1-20. MR 2491778, 10.4064/fm203-1-1 |
| Reference:
|
[7] Michael, E.: Topologies on spaces of subsets.Trans. Amer. Math. Soc. 71 (1951), 152-182. Zbl 0043.37902, MR 0042109, 10.1090/S0002-9947-1951-0042109-4 |
| Reference:
|
[8] Mill, J. {van}, Wattel, E.: Selections and orderability.Proc. Amer. Math. Soc. 83 (1981), 601-605. MR 0627702, 10.1090/S0002-9939-1981-0627702-4 |
| Reference:
|
[9] Tamariz-Mascar{'u}a, A.: Continuous selections on spaces of continuous functions.Comment. Math. Univ. Carolin. 47 (2006), 641-660. MR 2337419 |
| . |
