Minimal usco maps, densely continuous forms and upper semi-continuous functions
Let \(X,Y\) be Hausdorff spaces. The set-valued mapping \(F:X\to Y\) is an USCO map if it is upper semi-continuous with nonempty compact values; a minimal USCO map is one that is not properly contained in any other USCO map from \(X\) to \(Y\). A single-valued map \(f:X\to Y\) is subcontinuous (resp. quasi-continuous) at \(x\in X\) provided for every \(X\)-net \((x_{\sigma})_{\sigma}\) converging to \(x\), \((f(x_{\sigma}))_{\sigma}\) has a converging subnet (resp. for every \(Y\)-open \(V\ni f(x)\) and \(X\)-open \(U\ni x\) there is a nonempty \(X\)-open \(W\) with \(f(W)\subseteq V\)); \(f\) is subcontinuous (resp. quasi-continuous), if it is so at every point of \(X\). The main results is as follows: if \(Y\) is regular, then a set-valued mapping \(F:X\to Y\) is minimally USCO if and only if there is a quasicontinuous, subcontinuous function \(f:X\to Y\) such that the closure of the graph of \(f\) in \(X\times Y\) is equal to the graph of \(F\). The set \(M(X)\) of real-valued minimal USCO maps is shown to be uniformly isomorphic to the space of upper semi-continuous, locally bounded quasi-continuous real-valued functions endowed with the uniformity of uniform convergence. Similar questions are investigated for densely continuous forms and modifications.
scientific article; zbMATH DE number 5566186