Connectedness and Contractibility of Solutions in Set Optimization Under Set Less Order Relations

This study establishes the topological properties of solution sets in set optimization, focusing on their connectedness and contractibility. Utilizing the arcwise convexity and lower semicontinuity characteristics derived from scalarization techniques, we initially prove the connectedness of solution sets containing both weak and standard approximate solutions. Furthermore, by employing nonlinear scalarization methods, we verify the contractibility of weak minimal solution sets in set optimization frameworks.