Pareto efficiency without topology

Given a vector optimization problem over spaces endowed with a topological linear structure, existence results for optima (efficient points) are known. Relying only on the linear structure, the set of properly efficient points from a convex set is proved to be nonempty and the sets of Proper efficient points and Pareto efficient points coincide, provided that the set of internal points picked from the corresponding cone is nonempty. This result is appealing since the scalarization of the vector optimization problem is valid without topological requirements. A As an important consequence, we provide the Second Welfare Theorem in vector lattices and especially in Lebesgue spaces holds without topology.