A New Topological Framework and Its Application to Well-Posedness in Set-Valued Optimization

In this paper, we introduce a topology on the power set P(Z) of a partially ordered normed space Z from which we derive a topological convergence on P(Z) along with new concepts of continuity and semicontinuity for set-valued mappings. Our goal is to propose an appropriate framework to address set optimization problems involving set relations based on a cone ordering. Taking advantage of this new setting, we establish several results regarding the well-posedness of set-valued optimization problems that are consistent with the state-of-the-art.