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A Functional Analytic Proof of a Selection Lemma

Published online by Cambridge University Press:  20 November 2018

L. W. Baggett  and
Arlan Ramsay
L. W. Baggett
Affiliation:
University of Colorado, Boulder, Colorado
Arlan Ramsay
Affiliation:
University of Colorado, Boulder, Colorado
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If r is a mapping of a set X onto a set Y, then a selection for r is a mapping S of Y into X for which r o s is the identity. Analogously, if F is a mapping of a set Y into the power set 2Z of a set Z, then a selection for F is a mapping f of Y into Z such that ƒ(y) is an element of F(y) for all y in Y. These two notions are formally equivalent: Given r mapping X onto Y, define F(y) = r–l(y) and Z = X. Conversely, given F mapping Y into 2Z, define X to be the subset of Y × Z consisting of the pairs (y, z) for which z belongs to F(y), and define r on X by r(y, z) = y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 441 - 448
Copyright
Copyright © Canadian Mathematical Society 1980

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