The Role of Countable Paracompactness for Continuous Selections Avoiding Extreme Points

Abstract

The role of countable paracompactness to obtain a (setvalued) selection avoiding extreme points is investigated. In particular, we prove the following: Let $X$ be a topological space, $Y$ a normed space and $\varphi$ a lower semicontinuous compact-and convexvalued mapping of $X$ to $Y$. If one of the following conditions is valid, then $\varphi$ admits a lower semicontinuous set-valued selection $\phi$ such that $\phi(x)$ is compact and convex, and each point of $\phi(x)$ is not an extreme point of $\varphi(x)$ for each $x \in X$; (1) the infimum of the set of all diameters of $\varphi(x)$ with $x \in X$ is positive, (2) X is countably paracompact and the cardinality of $\varphi(x)$ is more than one for each $x \in X$. We also give characterizations of some topological spaces in terms of (set-valued) selections avoiding extreme points.