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This note generalizes Berge's maximum theorem to noncompact image sets. It isalso 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'stheorem, that states semi-continuity of value functions. Here we explain thatthe notion of a $\K$-inf-compact function introduced there is applicable tometrizable topological spaces and to more general compactly generatedtopological spaces. For Hausdorff topological spaces we introduce the notion ofa $\K\N$-inf-compact function ($\N$ stands for "nets" in $\K$-inf-compactness),which coincides with $\K$-inf-compactness for compactly generated and, inparticular, for metrizable topological spaces.

Berge's Maximum Theorem for Noncompact Image Sets