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
We introduce lower and upper semi-continuity of a map to the Banach space () for an infinite cardinal . We prove that the following conditions (i), (ii) and (iii) on a -space are equivalent: (i) For every two maps : () such that is upper semi-continuous, is lower semi-continuous and , there exists a continuous map : , with . (ii) For every Banach space , with every lower semi-continuous set-valued mapping : admits a continuous selection, where is the set of all non-empty compact convex sets in Y. (iii) is normal and every locally finite family of subsets of , with , 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.
J. Math. Soc. Japan, Volume 55, Number 2 (2003), 499-521.
First available in Project Euclid: 3 October 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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.)
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