On a theorem of Choquet and Dolecki
The author clarifies and extends slightly the work of \textit{S. Dolecki}, and \textit{A. Lechicki} [C. R. Acad. Sci., Paris, Ser. I 293, 219-221 (1981; Zbl 0486.54003); J. Math. Anal. Appl. 88, 547-554 (1982; Zbl 0503.54023)] and \textit{R. W. Hansell}, \textit{J. E. Jayne}, the author and \textit{C. A. Rogers} [Math. Z. 189, 297-318 (1985; Zbl 0544.54016)]. Let X, Y be Hausdorff spaces and F:X\(\to Y\) an upper semicontinuous set- valued map. A subset K of F(x) is said to be a peak of F at x, if, for every open set V containing K, there exists a neighborhood U of x such that F(U)\(\setminus F(x)\subset V\). Criteria (``Choquet-Dolecki Theorems'') are given in order that F has the smallest possible peak. It turns out that in unexpectedly general situations an upper semicontinuous map F has, for every x in X, a peak which is the smallest possible at x and moreover compact.
scientific article; zbMATH DE number 4064024