Berge's Theorem for Noncompact Image Sets
  For an upper semi-continuous set-valued mapping from one topological space to another and for a lower semi-continuous function defined on the product of these spaces, Berge's theorem states lower semi-continuity of the minimum of this function taken over the image sets. It assumes that the image sets are compact. For Hausdorff topological spaces, this paper extends Berge's theorem to set-valued mappings with possible noncompact image sets and studies relevant properties of minima. 
Eugene A. Feinberg, Pavlo O. Kasyanov and Nina V. Zadoianchuk
math.GN