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Berge's Maximum Theorem for Noncompact Image Sets

This note generalizes Berge's maximum theorem to noncompact image sets. It is also clarifies the results from E.A. Feinberg, P.O. Kasyanov, N.V. Zadoianchuk, "Berge's theorem for noncompact image sets," J. Math. Anal. Appl. 397(1)(2013), pp. 255-259 on the extension to noncompact image sets of another Berge's theorem, that states semi-continuity of value functions. Here we explain that the notion of a $\K$-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 $\K\N$-inf-compact function ($\N$ stands for "nets" in $\K$-inf-compactness), which coincides with $\K$-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.