Generic extensions of finite-valued u.s.c. selections

As a rule, most of the classical Michael-type selection theorems are analogues and, in some respects, generalizations of ordinary extension theorems. In this paper we show that the existence of set-valued u.s.c. selections for l.s.c. mappings is not related to the “usual” mapping-extension problem for u.s.c. mappings. In view of that, the paper is especially devoted to a proper notion of extending u.s.c. mappings that agrees well with the existing selection results. On this base new selection theorems dealing with controlled u.s.c. “extensions” of partial u.s.c. selections are obtained. Possible applications are illustrated in the dimension theory of normal spaces.