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Based on the improvement set  E , this paper aims at investigating density and connectedness of the sets of  E -optimal points, weak  E -optimal points,  E -quasi-optimal points,  E -Benson proper optimal points,  E -super optimal points and  E -strictly optimal points. By virtue of the separation theorem for convex sets, we obtain the scalarizaiton results for the sets of  E -optimal points, weak  E -optimal points,  E -Benson proper optimal points,  E -super optimal points and  E -strictly optimal points. Then we make a new attempt to establish some density theorems for the sets of these optimal points. Finally, we investigate connectedness and arcwise connectedness of the sets of various notions of  E -optimality by using the scalarization method and the density theorems.

Density and connectedness of optimal points with respect to improvement sets