  In this paper, we propose criteria for unboundedness of the images of
set-valued mappings having closed graphs in Euclidean spaces. We focus on
mappings whose domains are non-closed or whose values are connected. These
criteria allow us to see structural properties of solutions in vector
optimization, where solution sets can be considered as the images of solution
mappings associated to specific scalarization methods. In particular, we prove
that if the domain of a certain solution mapping is non-closed, then the weak
Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we
demonstrate two criteria to ensure that if the weak Pareto solution set is
disconnected then each connected component is unbounded.
