| Title:
|
A reverse viewpoint on the upper semicontinuity of multivalued maps (English) |
| Author:
|
Fenille, Marcio Colombo |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
4 |
| Year:
|
2013 |
| Pages:
|
415-423 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a multivalued map $\varphi \colon Y\multimap (X,\tau )$ between topological spaces, the upper semifinite topology $\mathcal {A}(\tau )$ on the power set $\mathcal {A}(X)=\{A\subset X \colon A\neq \emptyset \}$ is such that $\varphi $ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\mathcal {A}(\tau ))$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma $ on $\mathcal {A}(X)$, we consider a natural topology $\mathcal {R}(\Gamma )$ on $X$, constructed from $\Gamma $ satisfying $\mathcal {R}(\Gamma )=\tau $ if $\Gamma =\mathcal {A}(\tau )$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi \colon Y\multimap (X,\mathcal {R}(\Gamma ))$ to be equivalent to the continuity of the singlevalued map $\varphi \colon Y\rightarrow (\mathcal {A}(X),\Gamma )$. (English) |
| Keyword:
|
multivalued map |
| Keyword:
|
power set |
| Keyword:
|
upper semicontinuity |
| Keyword:
|
upper semifinite topology |
| MSC:
|
54A10 |
| MSC:
|
54C60 |
| idZBL:
|
Zbl 06260042 |
| idMR:
|
MR3231096 |
| DOI:
|
10.21136/MB.2013.143514 |
| . |
| Date available:
|
2013-11-09T20:26:34Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143514 |
| . |
| Reference:
|
[1] Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings.Topological Fixed Point Theory and Its Applications 4, 2nd ed Springer, Dordrecht (2006). Zbl 1107.55001, MR 2238622 |
| Reference:
|
[2] Michael, E.: Topologies on spaces of subsets.Trans. Am. Math. Soc. 71 (1951), 152-182. Zbl 0043.37902, MR 0042109, 10.1090/S0002-9947-1951-0042109-4 |
| . |
