A topological completeness concept with applications to the open mapping theorem and the separation of convex sets

Abstract

We introduce a completeness concept for convex sets in locally convex vector spaces which is based on the topological notion of p-completeness (also weak α-favourability). Using purely topological methods, we then establish an open mapping theorem for convex multifunctions and a separation theorem for convex sets generalizing the Tuckey-Klee separation theorem. Finally, we indicate that our notion of completeness encompasses Jameson's CS-closedness for convex sets, which hereby is shown to be essentially a topological notion.