Continuity concepts for correspondences

Question

Consider two metric spaces (X,d) and (Y,d') and a correspondence F from X to Y. Does a topology on the power set of Y, P(Y) exists such that F is upper (resp. lower hemi- continuous) if and only if F is continuous as a map from X to P(Y)?

Answer 1

Yes, these are the so-called Vietoris topologies. The upper Vietoris topology has a subbase consisting sets of the form $\{F\in 2^Y\mid F\subseteq O\}$ with $O$ open and the lower Vietoris topology has a subbase consisting sets of the form $\{F\in 2^Y\mid F\cap O\neq\emptyset\}$ 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."