Relative interior and closure of the set of inner points

Abstract

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Very recently this concept was generalized to the one of inner points in the scope of vector spaces, which, among other things, allows to characterize the linear dimension of a vector space and also serves to provide an intrinsic characterization of linear manifolds that was not possible by using internal points. Inner points of convex sets can be seen as the affine internal points, that is, the internal points with respect to the affine hull. In this manuscript we continue this research line in the scope of topological vector spaces to study the topological structure of the inner points. First, we prove that every infinite dimensional vector space has a convex subset free of inner points which is dense in every vector topology the vector space is endowed with. Also, we find the existence of convex sets in which the set of inner points is not open in the relative topology. Following this line, we also characterize the closed convex sets for which the set of inner points is not empty and open in the relative topology. Finally, we find an example of a convex set whose set of inner points is not contained in the set of inner points of its closure.