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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.

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