Journal of the Mathematical Society of Japan

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

Valentin GUTEV, Haruto OHTA, and Kaori YAMAZAKI

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We introduce lower and upper semi-continuity of a map to the Banach space c0(λ) for an infinite cardinal λ. We prove that the following conditions (i), (ii) and (iii) on a T1-space X are equivalent: (i) For every two maps g,h : Xc0(λ) such that g is upper semi-continuous, h is lower semi-continuous and gh, there exists a continuous map f : Xc0(λ), with gfh. (ii) For every Banach space Y, with w(Y)λ, every lower semi-continuous set-valued mapping : XCc(Y) admits a continuous selection, where Cc(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|λ, 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

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

First available in Project Euclid: 3 October 2007

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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.)

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


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.

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