On relations approximated by continuous functions. (English) Zbl 0694.54018

Let X, Y be metric spaces and R a closed multivalued map from X to Y. Let a sequence \(\{f_ n\}\subset C(X,Y)\) be such that the graphs of the maps \(f_ n\) converge in the Hausdorff metric of the space \(X\times Y\) to the graph of R.
Main results: (1) If X is locally compact and Y is complete then R is upper semicontinuous and has nonempty compact values. (2) Let X be locally connected and Y be locally compact. If the values of R are nonempty and compact then they are connected.


54C60 Set-valued maps in general topology
46E99 Linear function spaces and their duals
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