## Journal of the Mathematical Society of Japan

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

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

Valentin GUTEV, Haruto OHTA, and Kaori YAMAZAKI

#### 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$ : $X\to {c}_{0}$($\lambda $) such that $g$ is upper semi-continuous, $h$ is lower semi-continuous and $g\le h$, there exists a continuous map $f$ : $X\to {c}_{0}\left(\lambda \right)$, with $g\le f\le h$. (ii) For every Banach space $Y$, with $w\left(Y\right)\le \lambda ,$ every lower semi-continuous set-valued mapping $\varnothing $ : $X\to {\mathcal{C}}_{c}\left(Y\right)$ admits a continuous selection, where ${\mathcal{C}}_{c}\left(Y\right)$ 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 $\left|F\right|\le \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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/jmsj/1191419128

**Mathematical Reviews number (MathSciNet)**

MR1961298

**Zentralblatt MATH identifier**

1039.54012

**Subjects**

Primary: 54C60: Set-valued maps [See also 26E25, 28B20, 47H04, 58C06]

Secondary: 54C65: Selections [See also 28B20] 46B25: Classical Banach spaces in the general theory 54D15: Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

**Keywords**

selection insertion Banach space semi-continuous collectionwise normal perfectly normal countably paracompact paracompact

#### 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