This article deals with considering stability properties of Paretominimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painlevé–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painlevé–Kuratowski and use it together with a domination property to analyze stability conditions for the reference problems.