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Keywords:
multivalued map; power set; upper semicontinuity; upper semifinite topology
multivalued map; power set; upper semicontinuity; upper semifinite topology
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 $\varphi \colon Y\rightarrow (\mathcal {A}(X),\Gamma )$.
References:
[1] Górniewicz, L.: Topological Fixed Point Theory of Multivalued Mappings. Topological Fixed Point Theory and Its Applications 4, 2nd ed Springer, Dordrecht (2006). MR 2238622 | Zbl 1107.55001
[2] Michael, E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71 (1951), 152-182. DOI 10.1090/S0002-9947-1951-0042109-4 | MR 0042109 | Zbl 0043.37902
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