Extensions by Means of Expansions and Selections

Abstract

We provide proper mapping-characterizations of some embedding-like properties weaker than P ! -embedding. For instance, we show that a subset A of a space X is U ! -embedded in X if and only if for every continuous map g: A ! Y into a Banach space Y of weight w(Y) r !, there exists a continuous set-valued mapping ' of X into the nonempty compact subsets of Y such that g is a selection for ' j A (i.e., g(x) 2 '(x) for every x 2 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 ' of X into the non-empty compact subsets of a Banach space Y, every continuous selection g: A ! Y for ' j A is continuously extendable to the whole of X. Combining both results we get the well-known mapping-characterization of P ! -embedding which makes more transparent the relation FP ! ¼ U ! þ C*_. Other weak components of P ! -embedding are described in terms of expansions and selections, possible applications are demonstrated as well. (2000): 54C20, 54C25, 54C45, 54C60, 54C65. Mathematics Subject Classifications