Characterization of, Reflexive Spaces ems of Continuous Approximate Selec for Metric Projections

Let (X, z) be a topological space, and (Y, d) a metric space. A mapping F: X -+ 2’ which associates with every x E X a non-empty subset F(x) OF Y is said to be lower semi-continuous (I.s.c.) (respectively, upper semi-continuous (u.s.c.)) if, for each open set % in Y, the set (X E X: F(x) n %f # @ > (respectively, the set (X E X: F(x) c @ )) is open in X A mapping f: X-a Y is a selection for F if, for each x E X, f(x) E F(x). One of the most celebrated results on the existence of continuous selections is the following theorem of Michael [ 1 I]: If X is a paracompact (e.g., metric) space and F: X-+ 2’ is 1.s.c. and has closed convex images, then F admits a continuous selection. The key step in the proof of this theorem is the construction of continuous s-approximate selections. For an arbitrary non-empty set A C_ Y and E > 0, let B,(A) denote the union of open bahs with radii equal to F and centers running over A. A mapping f: X -+ Y is called an &-approximate selection for F: X-r 2' if for each x in X~(X) E &(I;(x)). In [7] Deutsch and Kenderov introduced two continuity properties for multivalued mappings and identified topologically those mappings which admit continuous s-approximate selections. 59 CO21-9045189 $3.00