Lower Hemi-Continuity, Open Sections, and Convexity: Counter Examples in Infinite Dimensional Spaces

Abstract

A lower hemi-continuous correspondence with open and convex values in Rn must have open lower sections. This well- known fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Since there is an increasing number of economic models that use correspondences in an infinite-dimensional setting, it is important to know whether or not the above fact remains valid in such applications. The aim of this paper is to show that the above fact no longer holds when Rn is replaced with an infinite-dimensional space. This is accomplished by using the standard orthonormal base in a Hilbert space H to construct two correspondences with values in H equipped with the weak topology. The first correspondence is lower hemi-continuous with open and convex values but does not have open lower sections. The second is a lower hemi-continuous correspondence that fails to have an open graph despite having open and convex upper and lower sections. These counter-examples demonstrate that in an infinite-dimensional setting, it is no longer possible to rely on the geometric properties of a lower hemi-continuous map (the convexity of its sections) to establish the topological properties (open lower sections, open graph) needed in many economic applications.