## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 32, Number 2 (2008), 277-290.

### 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.

#### Article information

**Source**

Tsukuba J. Math., Volume 32, Number 2 (2008), 277-290.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496165229

**Digital Object Identifier**

doi:10.21099/tkbjm/1496165229

**Mathematical Reviews number (MathSciNet)**

MR2477980

**Zentralblatt MATH identifier**

1162.54005

#### Citation

Yamauchi, Takamitsu. The Role of Countable Paracompactness for Continuous Selections Avoiding Extreme Points. Tsukuba J. Math. 32 (2008), no. 2, 277--290. doi:10.21099/tkbjm/1496165229. https://projecteuclid.org/euclid.tkbjm/1496165229