A Functional Analytic Proof of a Selection Lemma

Abstract

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.