Selections and topological convexity
A pair \((Y,\mathcal{C})\) is called a convex structure if \(Y\) is a nonempty set and \(\mathcal{C}\) is a family of subsets of \(Y\) satisfying the following conditions: (i) \(\emptyset, Y\in \mathcal{C}\); (ii) \(\bigcap \mathcal{D}\in \mathcal{C}\) for every nonempty \(\mathcal{D}\subset \mathcal{C}\); (iii) \(\bigcup \mathcal{D}\in \mathcal{C}\) for every nonempty \(\mathcal{D}\subset \mathcal{C}\) which is totally ordered by inclusion. The members of \(\mathcal{C}\) are called convex sets. The convex hull of a finite set \(F\subset Y\), that is \(\bigcap \{ C\in \mathcal{C}: F\subset C\}\), is called a polytope. A half-space \(H\) of \(Y\) is a convex subset \(H\) of \(Y\) such that \(Y\setminus H\) is also convex. A convex structure \((Y,\mathcal{C})\) is called an \(\mathbf{S_4}\) convex structure if for any two disjoint convex sets \(C, D\) there exists a half-space \(H\) such that \(C\subset H\) and \(D\subset Y\setminus H\). The author gives a simple geometrical proof for the following selection theorem due to \textit{M. van de Vel} [Trans. Am. Math. Soc. 336, No. 2, 463--496 (1993; Zbl 0781.52002)]: Let \(X\) be a paracompact space, \(Y\) be a metrizable \(\mathbf{S_4}\) convex structure with compact polytopes and connected convex sets, and let \(d\) be a compatible metric on \(Y\). Then each lower semicontinuous mapping \(\phi:X\rightsquigarrow Y\) with convex and \(d\)-complete values admits a single-valued continuous selection. The technique of ``small selections'' developed in the proof is then used to give also a direct simple proof of the classical Michael selection theorem in Fréchet spaces, and the Horvath selection theorem in metric l.c.-spaces.
scientific article; zbMATH DE number 5275704
scientific article