Bergeʼs maximum theorem for noncompact image sets

This note generalizes Bergeʼs maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Bergeʼs theorem, that states semi-continuity of value functions. Here we explain that the notion of a KK-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KNKN-inf-compact function (NN stands for “nets” in KK-inf-compactness), which coincides with KK-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.