## Journal of the Mathematical Society of Japan

### Selections and sandwich-like properties via semi-continuous Banach-valued functions

#### Abstract

We introduce lower and upper semi-continuity of a map to the Banach space $c_{0}$($\lambda$) for an infinite cardinal $\lambda$. We prove that the following conditions (i), (ii) and (iii) on a $T_{1}$-space $X$ are equivalent: (i) For every two maps $g,$$h$ : $X\rightarrow c_{0}$($\lambda$) such that $g$ is upper semi-continuous, $h$ is lower semi-continuous and $g\leq h$, there exists a continuous map $f$ : $X\rightarrow c_{0}(\lambda)$, with $g\leq f\leq h$. (ii) For every Banach space $Y$, with $w(Y)\leq/\%,$ every lower semi-continuous set-valued mapping $\emptyset$ : $X\rightarrow \mathscr{C}_{c}(Y)$ admits a continuous selection, where $\mathscr{C}_{c}(Y)$ is the set of all non-empty compact convex sets in Y. (iii)$X$ is normal and every locally finite family $F$ of subsets of $X$, with $|F|\leq\lambda$, has a locally finite open expansion provided it has a point-finite open expansion. We also characterize several paracompact-like properties by inserting continuous maps between semi-continuous Banach-valued functions.

#### Article information

Source
J. Math. Soc. Japan, Volume 55, Number 2 (2003), 499-521.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191419128

Digital Object Identifier
doi:10.2969/jmsj/1191419128

Mathematical Reviews number (MathSciNet)
MR1961298

Zentralblatt MATH identifier
1039.54012

#### Citation

GUTEV, Valentin; OHTA, Haruto; YAMAZAKI, Kaori. Selections and sandwich-like properties via semi-continuous Banach-valued functions. J. Math. Soc. Japan 55 (2003), no. 2, 499--521. doi:10.2969/jmsj/1191419128. https://projecteuclid.org/euclid.jmsj/1191419128