Extensions by Means of Expansions and Selections

Abstract

We provide proper mapping-characterizations of some embedding-like properties weaker than $P^{\lambda }$-embedding. For instance, we show that a subset A of a space X is $U^{\lambda }$-embedded in X if and only if for every continuous map g: A → Y into a Banach space Y of weight w(Y) ⩽ λ, there exists a continuous set-valued mapping φ of X into the nonempty compact subsets of Y such that g is a selection for φ∣A (i.e., g(x) ∈ φ(x) for every x ∈ A). On the other hand, we show that a subset A is C*-embedded in X if and only if for every continuous set-valued mapping φ of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A → Y for φ∣A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of $P^{\lambda }$-embedding which makes more transparent the relation ‘$P^{\lambda } = U^{\lambda } + C^{*}$’. Other weak components of $P^{\lambda }$-embedding are described in terms of expansions and selections, possible applications are demonstrated as well.