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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:
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


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