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Spaces of densely continuous forms. (English) Zbl 0892.54009
For each function $$f$$ from a topological space $$X$$ to a topological space $$Y$$ with a dense set $$D$$ of the points of continuity denote by $$\overline f$$ the closure of $$f|_D$$ in $$X\times Y$$. Then $$\overline f$$ is called a densely continuous form; it is a kind of multifunction from $$X$$ to $$Y$$. The paper studies mainly the topology of uniform convergence on compact sets on the space of densely continuous forms (which is denoted by $$D_k(X,Y))$$, for locally compact spaces $$X$$ and metric spaces $$Y$$ having a metric satisfying the Heine-Borel property. Under these conditions on $$X$$ and $$Y$$ the complete metrizability, metrizability and first countability of $$D_k(X,Y)$$ are equivalent to the condition that $$X$$ is $$G$$-compact. Also necessary and sufficient conditions are given for a subset of $$D_k(X,Y)$$ to be compact.

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