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where Ui, , Un are members of r. Let F: X-* Y be a set-valued function on X into Y such that F(x) is closed for each xeX. Then F induces a single-valued function f: X-*S( Y) on X into S( Y) by f (x) = F(x). Then F is continuous if and only if f is continuous with respect to the finite topology for S(Y). Let 8 be a collection of subsets of X. The set 8 generates a topology for X by: If A CX, a point xCX is an S-limit point for A in case every open set containing x contains a member of 8 which meets both { x } and A { x; see Young [2 ]. The topology generated by 8 is called the S-topology. We shall obtain a generalization of a theorem of Young [2] on the continuity of functions under changes of topology to include multivalued functions with each set F(x) finite. This result is then applied to derive a fixed-point theorem for such function1s. In this paper we shall assume that all spaces ar-e F-lausdorff.

Changes of topology and fixed points for multi-valued functions