Tsukuba Journal of Mathematics

The Role of Countable Paracompactness for Continuous Selections Avoiding Extreme Points

Takamitsu Yamauchi

Full-text: Open access

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


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