Selections and sandwich-like properties via semi-continuous Banach-valued functions.
The classical Katetov-Tong insertion theorem says that a \(T_1\) space \(X\) is normal if and only if for every two functions \(g,h: X\to\mathbb R\) such that \(g\) is upper semicontinuous, \(h\) is lower semicontinuous and \(g\leq h\), there exists a continuous function \(f: X\to\mathbb R\) such that \(g\leq f\leq h\). In this paper the authors obtain the natural generalization that arises when the real line is replaced by the Banach space \(c_0(\lambda)\), where \(\lambda\) is an infinite cardinal and \(c_0(\lambda)\) is defined as the set of all functions from \(\mathbb R^\lambda\) to \(\mathbb R\) such that, for each \(\varepsilon> 0\), the set \(\{\alpha< \lambda:| s(\alpha)|\geq \varepsilon\}\) is finite. They prove that the corresponding statement is equivalent to the fact that, for every Banach space \(Y\), with \(\omega(Y)\leq \lambda\), every lower semicontinuous set-valued mapping \(\varphi: X\to{\mathcal C}_C(Y)\) admits a continuous selection, where \({\mathcal C}_C(Y)\) is the set of all nonempty compact convex sets in \(Y\). Results characterizing paracompact-like properties in terms of sandwich-like statements and selections are shown and some interesting questions are included.
scientific article; zbMATH DE number 1943322