Finite-valued multiselections

It is shown that a connected space X is weakly orderable provided it has a finite-valued Vietoris continuous multiselection for its hyperspace F(X) of nonempty closed subsets. In fact, for connected spaces, every such multiselection is at most two-point valued, and X is compact whenever the multiselection is not singleton-valued at some element of F(X). Complementary to this result is a characterisation of weak orderability of connected spaces in terms of “proper” Vietoris continuous multiselections for hyperspaces of finite sets.