Topological convexities, selections and fixed points
The author introduces a convexity on a set \(X\), which is an algebraic closure operator \(A\mapsto [\![A]\!]\) from the power set \(\mathcal P(X)\) to itself such that \([\![\{x\}]\!]=\{x\}\) for all \(x\in X\) or, equivalently, a family \(\mathcal C\) of subsets of \(X\) which contains the whole space and the empty set as well as singletons and which is closed under arbitrary intersections and updirected unions. Uniform convex spaces, that is uniform topological spaces endowed with a convexity for which the convex hull operator is uniformly continuous, are also introduced in this paper. The definition is slightly different from that given by \textit{M. Van de Vel} [Trans. Am. Math. Soc. 336, No. 2, 463--496 (1993; Zbl 0781.52002)]. With these new concepts, the author obtains remarkable generalizations of very important results on selections and fixed points as follows: Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem \textit{à la Michael} holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions. The paper ends with a few examples of uniform convex spaces.
scientific article; zbMATH DE number 5275706
scientific article