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On approximation of multifunctions with respect to the Vietoris topology. (English) Zbl 0705.54016

The author considers approximation of multifunctions by step multifunctions with respect to the Vietoris topology. Previously G. Beer [Pac. J. Math. 87, 11-19 (1980; Zbl 0442.54014)] and the author [Ann. Soc. Math. Pol., Ser. 1, Commentat. Math. 25, 363-371 (1985; Zbl 0604.41020)] studied such approximation with respect to the Hausdorff metric or the Hausdorff uniformity. For a (continuous, upper semicontinuous, lower semicontinuous) multifunction F from a separable metric space X to topological space Y, theorems are proved that imply that F can be approximated by a sequence of upper semicontinuous and lower semicontinuous functions, respectively, with various conditions on the space Y. An example is given to show that conditions on the space Y are necessary for some of the theorems.
Reviewer: J.E.Keesling

MSC:

54C60 Set-valued maps in general topology
54B20 Hyperspaces in general topology
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