Inherent compactness of upper continuous set valued maps
The authors build on results of \textit{G. Choquet} [Ann. Univ. Grenoble, Sect. Sci. Math. Phys., II. Ser. 23, 57--112 (1948; Zbl 0031.28101)], \textit{I. A. Vainstein} [Dokl. Akad. Nauk SSSR, II. Ser. 57, 319--321 (1947; Zbl 0029.07702)] and \textit{S. Dolecki} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 25, 863--867 (1977; Zbl 0372.54008)], as extended in the first author's Ph. D. thesis [Univ. Mississippi 2006] and work involving the second author [for example in Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 45, No.~2, 225--235 (2005; Zbl 1102.54020)], and with \textit{R. Hansell, J. Jayne} and \textit{C. A. Rogers} in [Math. Z. 189, 297--318 (1985; Zbl 0544.54016)]. They aim to demonstrate a unity of method underlying the deduction of compactness of certain subsets derived from the upper semi-continuity of a set-valued map. Much of the paper is given over to a historical survey of the area, which gives appropriate motivation for the importance of what they call `Vaĭnsteĭn-Choquet Dolecki theorems' in demonstrating their thesis.
scientific article; zbMATH DE number 5566182
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