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Spaces of densely continuous forms, USCO and minimal USCO maps. (English) Zbl 1022.54012
The present paper is a study on topologies and corresponding convergences of set-valued maps, putting an emphasis on the topology of uniform convergence on compact sets on the space of densely continuous forms, upper semicontinuous (usco) and minimal upper semicontinuous maps. The author generalizes, completes and improves results by Hammer and McCoy which concern the space $$D(X,Y)$$ of densely continuous forms from $$X$$ to $$Y$$. Some of the main results are the following: For a Hausdorff topological space $$X$$ and a metric space $$(Y,d)$$, the topology $$D_k(X,Y)$$ of uniform convergence on compact sets on the space $$D(X,Y)$$ is metrizable iff $$D_k(X,Y)$$ is first countable iff $$X$$ is hemicompact. Moreover, if $$X$$ is a Hausdorff locally compact hemicompact space and $$(Y,d)$$ is a locally compact complete metric space, then $$D_k(X,Y)$$ is completely metrizable. Finally, the author studies a question posed by Hammer and McCoy (1998) whether two compatible metrics on $$Y$$ generate the same topologies of uniform convergence on compact sets on $$D(X,Y)$$ and discusses the completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps.

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