Uniform convergence on spaces of multifunctions. (English) Zbl 1164.54014

The author studies some topological properties of spaces of multifunctions equipped with the topology of uniform convergence. Namely, the multifunctions with closed graph \(G(X,2^X)\), USCO multifunctions \(U(X,Y)\), minimal USCO multifunctions \(M(X,Y)\), densely continuous forms \(D(X,Y)\) and locally bounded densely continuous forms \(D^*(X,Y)\) are investigated as subsets of the space of multifunctions with closed values \(F(X,2^X)\) equipped with the Hausdorff metric. By \(G(X,2^X)\), locally compact and locally countable compact spaces are characterized. The author uses the notion of active boundary for characterization of multifunctions with closed graph. More precisely, a multifunction \(F\) between two Hausdorff topological spaces has closed graph iff it has closed values and any value contains an active boundary. Most results of the paper concern conditions under which the sets mentioned above form a complete metric space.


54C35 Function spaces in general topology
54C60 Set-valued maps in general topology
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