Generalized metric properties and kernels of set-valued maps
Let \(F: X\to Y\) be a multifunction between topological spaces \(X\) and \(Y\). A subset \(K\subset X\) is called a kernel of \(F\) at \(x\in X\) if for each open set \(V\supset K\) there is a neighbourhood \(N\ni x\) with the property \(F(N)\subset V\cup F(x)\). The Choquet-Dolecki theorem states that if \(X\) is first-countable and \(Y\) is metrizable, then any upper semicontinuous \(F\) has a compact kernel at every \(x\in X\). The present paper proves that this theorem remains true if \(X\) is a \(\mathcal G\)-space and \(Y\) a \(k\)-semistratifiable one. Topological game theory is heavily used in the proof.
scientific article; zbMATH DE number 2136494