Abstract:Every l.s.c. mapping from a paracompact space into the non-empty, closed, convex subsets of a (not necessarily convex) $G_\delta $-subset of a Banach space admits a single-valued continuous selection provided every such mapping admits a convex-valued usco selection. This leads us to some new partial solutions of a problem raised by E. Michael.
Keywords: set-valued mapping, lower semi-continuous, upper semi-continuous, selection, countable-dimensional space
AMS Subject Classification: Primary 54C60, 54C65; Secondary 54F45