Adherence of minimizers for dual convergences

It is proved that many known convergences (e.g., continuous convergence, Isbell topology, compact-open topology, point-wise convergence) on the space of continuous maps (valued in a topological space) can be represented as the dual convergences with respect to collections of families of sets, and that they can be characterized in terms of the corresponding hyperspace convergences of the inverse images of closed sets. As a result, the convergence of real-valued functions for a dual convergence implies the convergence of their sets of minima on the corresponding hyperspace.