Yes, these are the so-called *Vietoris topologies*. The *upper Vietoris topology* has a subbase consisting sets of the form $\{F\in 2^Y\mid Y\subseteq O\}$ with $O$ open and the *lower Vietoris topology* has a subbase consisting sets of the form $\{F\in 2^Y\mid Y\cap O\neq\emptyset\}$ with $O$ open. The correspondence is then upper hemicontinuous if and only if the corresponding function is continuous under the upper Vietoris topology, and the correspondence is lower hemicontinuous if and only if the corresponding function is continuous under the lower Vietoris topology. The right keyword for finding more on the subject is "hyperspace topologies."