A STUDY OF MULTIVALUE FUNCTIONS

Abstract

The properties of multivalued functions have been studied independently by several authors; Smithson [5], whyburn [9], Ponomarev [10, 11] and others. However the concepts and the theorems which they had presented have a slight differences. In this paper, we attempt to systemize these differences and give them a uniformized view thereafter. In particular, as a main theorem we obtain a generalized form of Urysohn's lemma concerning upper semi-continuous (or lower semi-continuous) function. Because a continuous multivalued function is a generalization of a continuous single-valued function, many of the results on continuous multivalued functions are generalized form of continuous single-valued functions. However, since upper semi-continuity (or lower semi-continuity) is a weakened form of continuity of single-valued function, it can be easily guessed that some stronger conditions other than normality will be required in order to satisfy the Urysohn's lemma. We show that stratifiability of a space satisfies the above necessity. The theorem is as follows : Let A and B be disjoint closed subsets of a stratifiable space X. Then there exists a USC-function F:X→[0, 1] such that F[A] = 0 and F[B] = 1.