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For a multivalued map $\varphi :Y⊸(X,\tau )$ between topological spaces, the upper semifinite topology $𝒜\left(\tau \right)$ on the power set $𝒜\left(X\right)=\{A\subset X:A\ne \varnothing \}$ is such that $\varphi$ is upper semicontinuous if and only if it is continuous when viewed as a singlevalued map $\varphi :Y\to \left(𝒜\right(X),𝒜(\tau \left)\right)$. In this paper, we seek a result like this from a reverse viewpoint, namely, given a set $X$ and a topology $\Gamma$ on $𝒜\left(X\right)$, we consider a natural topology $ℛ\left(\Gamma \right)$ on $X$, constructed from $\Gamma$ satisfying $ℛ\left(\Gamma \right)=\tau$ if $\Gamma =𝒜\left(\tau \right)$, and we give necessary and sufficient conditions to the upper semicontinuity of a multivalued map $\varphi :Y⊸(X,ℛ(\Gamma \left)\right)$...